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Mircea Mustata & Mihnea PopaHodge ideals for Q-divisors: birational approachTome 6 (2019), p. 283-328.

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Tome 6, 2019, p. 283–328 DOI: 10.5802/jep.94

HODGE IDEALS FOR Q-DIVISORS:

BIRATIONAL APPROACH

by Mircea Mustata & Mihnea Popa

Abstract. — We develop the theory of Hodge ideals for Q-divisors by means of log resolutions,extending our previous work on reduced hypersurfaces. We prove local (non-)triviality criteriaand a global vanishing theorem, as well as other analogues of standard results from the theoryof multiplier ideals, and we derive a new local vanishing theorem. The connection with theV -filtration is analyzed in a sequel.

Résumé (Idéaux de Hodge pour des Q-diviseurs : approche birationnelle)Nous développons la théorie des idéaux de Hodge pour les Q-diviseurs à l’aide de log réso-

lutions, généralisant notre précédent travail sur les hypersurfaces réduites. Nous obtenons descritères de (non) trivialité locale et un théorème d’annulation global, ainsi que d’autres ana-logues de résultats standard de la théorie des idéaux multiplicateurs, et nous en déduisons unnouveau théorème d’annulation local. Nous analysons la relation avec la V -filtration dans unautre article.

Contents

A. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283B. Hodge ideals via log resolutions, and first properties. . . . . . . . . . . . . . . . . . . . . . . . . 287C. Local study and global vanishing theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309D. Restriction, subadditivity, and semicontinuity theorems. . . . . . . . . . . . . . . . . . . . . . 323References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

A. Introduction

In this paper we continue the study of Hodge ideals initiated in [MP16], [MP18b],by considering an analogous theory for arbitraryQ-divisors. The emphasis here is on abirational definition and study of Hodge ideals, while the companion paper [MP18a] isdevoted to a study based on their connection with the V -filtration, inspired by [Sai16].

2010 Mathematics Subject Classification. — 14F10, 14J17, 32S25, 14F17.Keywords. — Hodge ideals, D-modules, Hodge filtration, vanishing theorems, minimal exponent.

MM was partially supported by NSF grant DMS-1701622; MP was partially supported by NSF grantDMS-1700819.

e-ISSN: 2270-518X http://jep.centre-mersenne.org/

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284 M. Mustata & M. Popa

Both approaches turn out to provide crucial information towards a complete under-standing of these objects.

Let X be a smooth complex variety. If D is reduced divisor on X, the Hodge idealsIk(D), with k > 0, are defined in terms of the Hodge filtration on the DX -moduleOX(∗D) of functions with poles of arbitrary order along D. Indeed, this DX -moduleunderlies a mixed Hodge module on X, and therefore comes with a Hodge filtrationF•OX(∗D), which satisfies

FkOX(∗D) = Ik(D)⊗ OX((k + 1)D

)for all k > 0.

See [MP16] for details, and for an extensive study of the ideals Ik(D).Our goal here is to provide a similar construction and study in the general case.

A natural device for dealing with the fact that fractional divisors are not directlyrelated to Hodge theory is to use new objects derived from covering constructions.Let D be an arbitrary effective Q-divisor on X. Locally, we can write D = αH, forsome α ∈ Q>0 and H = div(h), the divisor of a nonzero regular function; we alsodenote by Z the support of D. A well-known construction associates to this data atwisted version of the localization D-module above, namely

M (h−α) := OX(∗Z)h−α,

that is the rank 1 free OX(∗Z)-module with generator the symbol h−α, on which aderivation D of OX acts by

D(wh−α) :=(D(w)− αwD(h)

h

)h−α.

It turns out that this DX -module can be endowed with a natural filtration FkM (h−α),with k > 0, which makes it a filtered direct summand of a D-module underlying amixed Hodge module on X; see Section 2. This plays a role analogous to the Hodge fil-tration, and just as in the reduced case one can show that FkM (h−α) ⊆ OX(kZ)h−α.This is done in Section 3 and Section 4, by first analyzing the case when Z is a smoothdivisor (in this case, if dDe = Z, then the inclusion is in fact an equality). It is there-fore natural to define the k-th Hodge ideal of D by the formula

FkM (h−α) = Ik(D)⊗OX OX(kZ)h−α.

Similarly to [MP16], one of our main goals here is to study Hodge ideals ofQ-divisors by means of log resolutions. To this end, let f : Y → X be a log resolutionof the pair (X,D) that is an isomorphism over U = X r Z, and denote g = h f .There is a filtered isomorphism(

M (h−α), F)' f+

(M (g−α), F

).

Denoting G = f∗D and E = Supp(G), so that E is a simple normal crossing divisor,it turns out that there exists a complex on Y :

C•g−α(−dGe) : 0 −→ OY (−dGe)⊗OY DY −→ OY (−dGe)⊗OY Ω1Y (logE)⊗OY DY

−→ · · · −→ OY (−dGe)⊗OY ωY (E)⊗OY DY −→ 0,

J.É.P. — M., 2019, tome 6

Hodge ideals for Q-divisors: birational approach 285

which is placed in degrees −n, . . . , 0, whose differential is described in Section 6. Thiscomplex has a natural filtration given, for k > 0, by subcomplexes

Fk−nC•g−α(−dGe) := 0→OY (−dGe)⊗Fk−nDY →OY (−dGe)⊗Ω1

Y (logE)⊗Fk−n+1DY

→ · · · → OY (−dGe)⊗ ωY (E)⊗ FkDY → 0.

Extending [MP16, Prop. 3.1], we show in Proposition 6.1 and Proposition 7.1 thatthere is a filtered quasi-isomorphism(

C•g−α(−dGe), F)'(Mr(g

−α), F),

where Mr(g−α) is the filtered right DY -module associated to M (g−α). Thus one can

use(C•g−α(−dGe), F

)as a concrete representative for computing the filtered D-module

pushforward of(Mr(g

−α), F), hence for computing the ideals Ik(D). More precisely,

we haveR0f∗Fk−n

(C•g−α(−dGe)⊗DY DY→X

)' h−αωX(kZ)⊗OX Ik(D).

See Theorem 8.1 for a complete picture regarding this push-forward operation.This fact, together with special properties of the filtration on D-modules underlying

mixed Hodge modules, leads to our main results on Hodge ideals, which are collectedin the following:

Theorem A. — In the set-up above, the Hodge ideals Ik(D) satisfy:(i) I0(D) is the multiplier ideal I

((1− ε)D

), so in particular I0(D) = OX if and

only if the pair (X,D) is log canonical; see Section 9.(ii) If Z has simple normal crossings, then

Ik(D) = Ik(Z)⊗ OX(Z − dDe),

while Ik(Z) can be computed explicitly as in [MP16, Prop. 8.2]; see Section 7. In par-ticular, if Z is smooth, then Ik(D) = OX(Z − dDe) for all k; cf. also Corollary 11.12.

(iii) The Hodge filtration is generated at level n− 1, where n = dimX, i.e.,

F`DX ·(Ik(D)⊗ OX(kZ)h−α

)= Ik+`(D)⊗ OX

((k + `)Z

)h−α

for all k > n− 1 and ` > 0; see Section 10.(iv) There are non-triviality criteria for Ik(D) at a point x ∈ D in terms of the

multiplicity of D at x; see Section 11.(v) If X is projective, Ik(D) satisfy a vanishing theorem analogous to Nadel Van-

ishing for multiplier ideals; see Section 12.(vi) If Y is a smooth divisor in X such that Z|Y is reduced, then Ik(D) satisfy

Ik(D|Y ) ⊆ Ik(D) · OY ,

with equality when Y is general; see Section 13 for a more general statement.(vii) If X → T is a smooth family with a section s : T → X, and D is a rela-

tive divisor on X that satisfies a suitable condition (see Section 14 for the precisestatement) then

t ∈ T | Ik(Dt) 6⊆ mqs(t)is an open subset of T , for each q > 1.

J.É.P. — M., 2019, tome 6


(viii) If D1 and D2 are Q-divisors with supports Z1 and Z2, such that Z1 + Z2 isalso reduced, then the subadditivity property

Ik(D1 +D2) ⊆ Ik(D1) · Ik(D2)

holds; see Section 15 for a more general statement.

For comparison, the list of properties of Hodge ideals in the case when D is reducedis summarized in [Pop19, §4]. While much of the story carries over to the setting ofQ-divisors – besides of course the connection with the classical Hodge theory of thecomplement U = XrD, which only makes sense in the reduced case – there are a fewsignificant points where the picture becomes more intricate. For instance, the boundsfor the generation level of the Hodge filtration can become worse. Moreover, we donot know whether the inclusions Ik(D) ⊆ Ik−1(D) continue to hold for arbitraryQ-divisors. New phenomena appear as well: unlike in the case of multiplier ideals,for rational numbers α1 < α2, usually the ideals Ik(α1Z) and Ik(α2Z) cannot becompared for k > 1; see for instance Example 10.5.

It turns out however that most of these issues disappear if one works modulo theideal of the hypersurface, at least for rational multiples of a reduced divisor. This,as well as other basic facts, is addressed in the sequel [MP18a], which studies Hodgeideals from a somewhat different point of view, namely by comparing them to the(microlocal) V -filtration induced on OX by h. This is inspired by the work of Saito[Sai16] in the reduced case. In the statement below we summarize some of theseproperties, which complement the results in Theorem A, but which we do not knowhow to obtain with the methods of this paper.

Theorem B ([MP18a]). — Let D = αZ, where Z is a reduced divisor and α ∈ Q>0.Then the following hold:

(1) Ik(D) + OX(−Z) ⊆ Ik−1(D) + OX(−Z) for all k.(2) If α ∈ (0, 1], then Ik(D) = OX ⇐⇒ k 6 αZ − α, where αZ is the negative of

the largest root of the reduced Bernstein-Sato polynomial of Z.(3) If Ik−1(D) = OX (we say that (X,D) is (k−1)-log canonical), then Ik+1(D) ⊆

Ik(D).(4) Fixing k, there exists a finite set of rational numbers 0 = c0 < c1 < · · · < cs <

cs+1 = 1 such that for each 0 6 i 6 s and each α ∈ (ci, ci+1] we have

Ik(αZ) · OZ = Ik(ci+1Z) · OZ = constant

and such thatIk(ci+1Z) · OZ ( Ik(ciZ) · OZ .

Going back to the description of Hodge ideals by means of log resolutions, thestrictness of the Hodge filtration for the push-forwards of (summands of) mixed Hodgemodules leads to the following local Nakano-type vanishing result for Q-divisors:

J.É.P. — M., 2019, tome 6


Corollary C. — Let D be an effective Q-divisor on a smooth variety X of dimen-sion n, and let f : Y → X be a log resolution of (X,D) that is an isomorphism overX r Supp(D). If E = (f∗D)red, then

Rqf∗(ΩpY (logE)⊗OY OY (−df∗De)

)= 0 for p+ q > n.

Note that for p = n this is the local vanishing for multiplier ideals [Laz04, Th. 9.4.1],since E−df∗De = −[(1−ε)f∗D] for 0 < ε 1. In general, the statement extends thecase of reduced D in [Sai07, Cor. 3] (cf. also [Sai16, §A.5]). Unlike [MP16, Th. 32.1]regarding that case, at the moment we are unable to prove this corollary via moreelementary methods.

A different series of applications, given in [MP18a], uses the results proved inthis paper together with the relationship between Hodge ideals of Q-divisors andthe V -filtration, in order to describe the behavior of the invariant αZ described inTheorem B (called the minimal exponent of Z). For instance, the triviality criterionproved here as Proposition 11.2 leads to a lower bound [MP18a, Cor.D] for αZ in termsof invariants on a log resolution, addressing a question of Lichtin and Kollár. Moreover,the results in Theorem A (vi) and (vii), and Corollary 11.11, lead to effective boundsand to restriction and semicontinuity statements for αZ , in analogy with well-knownproperties of log canonical thresholds; for details see [MP18a, §6].

B. Hodge ideals via log resolutions, and first properties

Let X be a smooth complex algebraic variety. Given an effective Q-divisor D on X,our goal is to attach to D ideal sheaves Ik(D) for k > 0; when D is a reduced divisor,these will coincide with the Hodge ideals in [MP16].

1. A brief review of Hodge modules. — A key ingredient for the definition of ourinvariants is Saito’s theory of mixed Hodge modules. In what follows, we give a briefpresentation of the relevant objects, and recall a few facts that we will need. Fordetails, we refer to [Sai90].

Given a smooth n-dimensional complex algebraic variety X, we denote by DX

the sheaf of differential operators on X. This carries the increasing filtration F•DX

by order of differential operators. A left or right D-module is a left, respectivelyright, DX -module, which is quasi-coherent as an OX -module. There is an equivalencebetween the categories of left and right D-modules, which at the level of OX -modulesis given by

M −→ N := M ⊗OX ωX and N −→HomOX (ωX ,N ).

For example, this equivalence maps the left D-module OX to the right D-module ωX .For a thorough introduction to the theory of D-modules, we refer to [HTT08].

A filtered left (or right) D-module is a D-module M , together with an increasingfiltration F = F•M that is compatible with the order filtration on DX and whichis good, in a sense to be defined momentarily. A morphism of filtered D-modules is

J.É.P. — M., 2019, tome 6


required to be compatible with the filtrations. The equivalence between left and rightD-modules extends to the categories of filtered modules, with the convention that

Fp−n(M ⊗OX ωX) = FpM ⊗OX ωX .

A filtration F•M on a coherent D-module M is good if the corresponding gradedobject grF• M :=

⊕k FkM /Fk−1M is locally finitely generated over grF• DX . We

note that every coherent D-module admits a good filtration, but this is far frombeing unique.

We now come to the key objects in Saito’s theory, the mixed Hodge modules from[Sai90]. Such an object is given by the data M = (M , F,P, ϕ,W ), where:

(i) (M , F ) is a filtered D-module, with M a holonomic left (or right) D-module,with regular singularities; F is the Hodge filtration of M .

(ii) P is a perverse sheaf of Q-vector spaces on X.(iii) ϕ is an isomorphism between PC = P ⊗Q C and DR(M ), i.e., the perverse

sheaf corresponding to M via the Riemann-Hilbert correspondence.(iv) W is a finite, increasing filtration on (M , F,P, ϕ), the weight filtration of the

mixed Hodge module.For a such an object to be a mixed Hodge module, it has to satisfy a complicated

set of conditions of an inductive nature, which we do not discuss here. The mainreference for the basic definitions and results of this theory is [Sai90]; see also [Sai17]for an introduction.

Given a mixed Hodge module (M , F,P, ϕ,W ), we say that the filtered D-module(M , F ) is a Hodge D-module (or that it underlies a mixed Hodge module). In fact,this is the only piece of information that we will be concerned with in this article. Thebasic example of a mixed Hodge module is QH

X [n], the trivial one. In this case, thefiltered D-module is the structure sheaf OX , with the filtration such that grFp OX = 0

for all p 6= 0. The corresponding perverse sheaf is QX [n] and the weight filtration issuch that grWp OX = 0 for p 6= n.

The mixed Hodge modules on X form an Abelian category, denoted MHM(X).Morphisms in this category are strict with respect to both the Hodge and the weightfiltration. The corresponding bounded derived category is denoted Db

(MHM(X)

).

Mixed Hodge modules satisfy Grothendieck’s 6 operations formalism. The relevantfact for us is that to every morphism f : X → Y of smooth complex algebraic varietieswe have a corresponding push-forward functor f+ : Db

(MHM(X)

)→ Db

(MHM(Y )

)(this is denoted by f∗ in [Sai90]). Moreover, if g : Y → Z is another such morphism,we have a functorial isomorphism (g f)+ ' g+ f+.

Regarding the push-forward functor for mixed Hodge modules, we note that on thelevel of D-modules, it coincides with the usual D-module push-forward. Moreover, iff : X → Y is proper and if we denote by FM(DX) the category of filtered D-moduleson X (here it is convenient to work with right D-modules), then Saito defined in[Sai88] a functor

f+ : Db(FM(DX)

)−→ Db

(FM(DY )

).

J.É.P. — M., 2019, tome 6


This is compatible with the usual direct image functor for right D-modules and itis used to define the push-forward between the derived categories of mixed Hodgemodules at the level of filtered complexes. With a slight abuse of notation, if (M , F )

underlies a mixed Hodge moduleM on X and if f : X → Y is an arbitrary morphism,then we write f+(M , F ) for the object in Db

(FM(DY )

)underlying f+M .

An important feature of the push-forward of Hodge D-modules with respect toproper morphisms is strictness. This says that if f : X → Y is proper and (M , F )

underlies a mixed Hodge module on X, then f+(M , F ) is strict as an object inDb(FM(DY )

)(and moreover, each Hif+(M , F ) underlies a Hodge DY -module). This

means that the natural mapping

(1.1) Rif∗(Fk(M

L⊗DX DX→Y )

)−→ Rif∗(M

L⊗DX DX→Y )

is injective for every i, k ∈ Z. Taking FkHif+(M , F ) to be the image of this map, weget the filtration on Hif+(M , F ).

The push-forward with respect to open embeddings is more subtle. For example,suppose that Z is an effective divisor on the smooth varietyX and j : U = XrZ → X

is the corresponding open immersion. Recall that OX(∗Z) is the push-forward j∗OU ;on a suitable affine open neighborhood V of a given point in X, this is given bylocalizing OX(V ) at an equation defining Z ∩ V in V . OX(∗Z) has a natural leftD-module structure induced by the canonical D-module structure on OX . In fact, assuch we have OX(∗Z) ' j+OU (in general, for a DU -module M , the D-module push-forward j+M agrees with j∗M , with the induced DX -module structure). We thussee that OX(∗Z) carries a canonical filtration such that the corresponding filteredD-module underlies j+QH

U [n]. This filtration is the one that leads to the Hodge idealsstudied in [MP16].

2. Filtered D-modules associated to Q-divisors. — Let X be a smooth complexalgebraic variety, with dim(X) = n. The ideals we associate to effective Q-divisorson X arise from certain Hodge D-modules. The D-modules themselves have beenextensively studied: these are the D-modules attached to rational powers of functionson X. We proceed to recall their definition.

Consider a nonzero h ∈ OX(X) and β ∈ Q. We denote by Z the reduced divisoron X with the same support as H = div(h) and let j : U = X r Supp(Z) → X bethe inclusion map. We consider the left DX -module M (hβ), which is a rank 1 freeOX(∗Z)-module with generator the symbol hβ , on which a derivation D of OX acts by

D(whβ) :=(D(w) + w

β ·D(h)

h

)hβ .

We will denote the corresponding right DX -module by Mr(hβ). This can be described

as hβωX(∗Z), an OX -module isomorphic to ωX(∗Z), and such that if x1, . . . , xn arelocal coordinates, then

(hβwdx1 · · · dxn)∂i = −hβ( ∂w∂xi

+ wβ

h· ∂h∂xi

)dx1 · · · dxn

for every i with 1 6 i 6 n.

J.É.P. — M., 2019, tome 6


Remark 2.1. — When β ∈ Z, we have a canonical isomorphism of left DX -modules

(2.1) M (hβ) ' OX(∗Z), whβ 7−→ w · hβ ,

where on the localization OX(∗Z) we consider the natural DX -module structure in-duced from OX . Note that OX(∗Z) is also the D-module push-forward j+OU .

Remark 2.2. — For every positive integer m, we have a canonical isomorphism of leftDX -modules

M (hβ) 'M((hm)β/m

), whβ 7−→ w(hm)β/m.

Remark 2.3. — We can define, more generally, left D-modules M (hβ1

1 · · ·hβrr ), fornonzero regular functions h1, . . . , hr ∈ OX(X) and rational numbers β1, . . . , βr. If ìare positive integers such that βi/ì = β for all i and if h =

∏i h

ìi , then we have an

isomorphism of left DX -modules

M (hβ1

1 · · ·hβrr ) 'M (hβ).

Remark 2.4. — If r is an integer, then we have an isomorphism of left DX -modules

M (hβ) −→M (hr+β), whβ 7−→ (wh−r)hr+β .

Let now D be an effective Q-divisor on X. We denote by Z the reduced divisorwith the same support as D. As above, we put U = X rZ and let j : U → X be theinclusion map. We first assume that we can write D = α · div(h) for some nonzeroh ∈ OX(X) and α ∈ Q>0 (this is of course always the case locally). To this data wecan associate the DX -module M (h−α); later it will be more convenient to considerequivalently (according to Remark 2.4) the DX -module M (h1−α). This depends onthe choice of h; however, if we replace h by hm and α by α/m, for some positiveinteger m, the D-module does not change (see Remark 2.2). In particular, we mayalways assume that α = 1/`, for a positive integer `.

Remark 2.5. — Suppose that D′ is a Q-divisor with the same support as D andsuch that D − D′ = div(u), for some u ∈ OX(X). Suppose that we can write D′ =

(1/`) · div(h′) for some h′ ∈ OX(X) and some positive integer `. In this case wecan also write D = (1/`) · div(h), where h = u`h′, and we have an isomorphism ofDX -modules

(2.2) M (h−1/`) −→M (h′−1/`

), gh−1/` 7−→ gu−1h′−1/`

.

Our first goal is to show that M (h−α) is canonically a filtered DX -module. Let `be a positive integer such that `α ∈ Z. Consider the finite étale map p : V → U , whereV = SpecOU [y]/(y` − h−`α). Note that this fits in a Cartesian diagram

(2.3)

V //

p

W

q

Uj// X,

J.É.P. — M., 2019, tome 6


in whichW = SpecOX [z]/(z` − h`α),

such that the map V →W pulls z back to y−1 = y`−1h`α.

Lemma 2.6. — We have an isomorphism of left DX-modules

(2.4) j+p+OV '`−1⊕i=0

M (h−iα),

with the convention that the first summand is OX(∗Z).

Proof. — Since p is finite étale, it follows that we have a canonical isomorphismτ : p∗DU ' DV , and for every DV -module M we have p+M ' p∗M , with the actionof DU induced via the isomorphism τ .

By mapping gyi to gh−iα, where g is a section of OX and 0 6 i 6 `− 1, we obtainan isomorphism of OX -modules as in (2.4). In order to see that this is an isomorphismof DX -modules, consider a local derivation D of OX and note that since y` = h−`α,by identifying D with its pull-back to V we have

D(yi) = iyi−1D(y) = −iαyi D(h)

h,

which via our map corresponds to D(h−iα). This implies the assertion.

It follows from the lemma that the right-hand side of (2.4) is the D-module cor-responding to the mixed Hodge module push-forward (j p)+QH

V [n]. In particular, itcarries a canonical structure of filtered D-module.

Remark 2.7. — Let us see what happens if we replace ` by a multiple m`. Letp` : V` → U and pm` : Vm` → U be the corresponding étale covers. Note that

Vm` = SpecOU [y]/(y`m − h−`mα)

decomposes as a disjoint union of m copies of V`, and thus we have an isomorphismof filtered DX -modules (and a corresponding isomorphism of mixed Hodge modules)

(2.5) j+(pm`)+OVm` '(j+(p`)+OV`

)⊕m.

If η is a primitive root of 1 of order `m, and if on each side of (2.5) we consider thedecompositions (2.4), then the isomorphism maps

h−iα 7−→(ηish−c`α · h−dα

)06s6m−1,

where we write i = `c+ d, with 0 6 c 6 m− 1 and 0 6 d 6 `− 1.

We can interpret the isomorphism in (2.4) in terms of a suitable µ`-action, where µ`is the group of `-th roots of 1 in C∗. Note that we have a natural action of µ` on Wsuch that via the corresponding action on OW , an element λ ∈ µ` maps zi to λizi.If we let µ` act trivially on X, then q is an equivariant morphism (in fact, q is thequotient morphism with respect to the µ`-action). It is clear that q−1(Z) is fixedby the µ`-action and we have an induced µ`-action on W r q−1(Z) = V . This inturn induces a µ`-action on j+p+OV and the isomorphism in (2.4) corresponds to

J.É.P. — M., 2019, tome 6


the isotypic decomposition of j+p+OV , such that every λ ∈ µ` acts on M (h−iα) bymultiplication with λ−i.

Lemma 2.8. — The filtration on j+p+OV is preserved by the µ`-action. Therefore wehave an induced filtration on each M (h−iα) such that (2.4) is an isomorphism offiltered D-modules.

Proof. — One way to see this is by using a suitable equivariant resolution of W .Let W ′ be the disjoint union of the irreducible components of W and q′ : W ′ → W

the canonical morphism. It is clear that the µ`-action on W induces an action on W ′such that q′ is equivariant. Since V is contained in the smooth locus of W , it has anopen immersion into W ′. We use equivariant resolution of singularities to constructa µ`-equivariant morphism ϕ : Y →W ′ that is an isomorphism over V and such that(q q′ ϕ)∗(Z) is a divisor with simple normal crossings. Let g = q q′ ϕ. If E is thereduced, effective divisor supported on g−1(Z), then we have an isomorphism of fil-tered D-modules (induced by a corresponding isomorphism of mixed Hodge modules)

(2.6) j+p+OV ' g+j+OV ' g+OY (∗E),

where j : Y r Supp(E) → Y is the inclusion map.We can deduce the assertion in the lemma from an explicit computation of the

filtration on j+p+OV via the isomorphism (2.6), as follows. First, since we deal withD-module push-forward, it is more convenient to work with right D-modules. We willthus compute g+ωY (∗E), where ωY (∗E) is the filtered right D-module correspondingto OY (∗E).

Since E is a simple normal crossing divisor, ωY (∗E) has a resolution by a com-plex C• of filtered right DY -modules

0 −→ C−n −→ · · · −→ C0 −→ 0,

where Ci = Ωi+nY (logE)⊗OY DY , with the filtration given by

Fk−nCi = Ωi+nY (logE)⊗OY Fk+iDY .

For a description of the maps in this complex, see the beginning of Section 6 below;a proof of the fact that it resolves ωY (∗E) is given in [MP16, Prop. 3.1]. We can thuscompute Fkg+ωY (∗E) as the image of the injective map

R0g∗(Fk(C• ⊗DY DY→X)

)−→ R0g∗(C

• ⊗DY DY→X) = g+ωY (∗E).

Since g is equivariant and the action of µ` on Y induces an action on E (in fact, itfixes E), the above description implies that each Fkg+ωY (∗E) is preserved by theµ`-action.

Remark 2.9. — We note that the filtration on j+p+OV induces the canonical filtrationon the first summand OX(∗Z). Indeed, on U we have a morphism of mixed Hodgemodules QH

U [n]→ p+QHV [n]. Applying j+ and only considering the underlying filtered

D-modules, we obtain a morphism j+OU → j+p+OV , which is an isomorphism ontothe first summand.

J.É.P. — M., 2019, tome 6


Definition 2.10. — Given α > 0, choose ` > 2 such that `α ∈ Z. In this case M (h−α)

appears as the second summand in the decomposition (2.4). We define the filtration

FkM (h−α) for k > 0

to be the filtration induced from the canonical filtration on j+p+OV . It is straightfor-ward to see, using the discussion in Remark 2.7 that this filtration does not changeif we replace ` by a multiple; therefore it is independent of `. Moreover, we note thatif α is an integer, using the same Remark 2.7, the isomorphism M (h−α) ' OX(∗Z)

is an isomorphism of filtered D-modules.

In this definition, a priori different covers have to be considered for each of thesummands M (h−iα). However, we have:

Lemma 2.11. — With the filtration defined above, the isomorphism (2.4) is an iso-morphism of filtered D-modules.

Proof. — By Lemma 2.8, we only need to show that for every i with 0 6 i 6 ` − 1,the filtration induced on M (h−iα) by that on j+p+OV coincides with the one givenin the above definition. For i = 0 this follows from Remark 2.9. If i > 0, consider thecover used to define the filtration on M (h−iα), namely

p′ : V ′ = SpecOU [y]/(y` − hiα`) −→ U.

Note that we have a finite morphism ψ : V → V ′ of varieties over U , that pulls-back y to yi. We have a canonical morphism of mixed Hodge modules QH

V ′ [n] →ψ+Q

HV [n]. Applying j+p′+ and passing to the underlying filtered D-modules, we obtain

a morphism of filtered D-modules j+p′+OV ′ → j+p+OV that is the identity on thesummand M (h−iα). This proves our claim.

Remark 2.12. — It is clear from definition that for every α > 0 and every positiveinteger m, the isomorphism

M (h−α) −→M((hm)−α/m

), ghm 7−→ g(hm)−α/m

is an isomorphism of filtered D-modules.

Remark 2.13. — In the setting of Remark 2.5, the isomorphism (2.2) is an isomor-phism of filtered DX -modules. This is clear if ` = 1, hence we assume ` > 2. Letp : V → U and p′ : V ′ → U be the canonical projections, where

V = SpecOU [y]/(y` − h) and V ′ = SpecOU [y]/(y` − h′).

We have an isomorphism ϕ : V ′ → V of schemes over U , where ϕ∗(y) = uy. Thisinduces an isomorphism of filtered DX -modules

j+p+OV ' j+p′+OV ′ ,

which via the identifications given by Lemma 2.6 is the direct sum`−1⊕i=0

M (h−i/`) '`−1⊕i=0

M (h′−i/`

)

of the isomorphisms (2.2). For i = 1, we obtain our assertion.

J.É.P. — M., 2019, tome 6


A special case of the above remark implies that for every α > 0 the isomorphism

M (h−α) −→M (h−α−1), gh−α 7−→ (gh)h−α−1

is an isomorphism of filtered D-modules. We use this to put a structure of filteredD-module on M (hβ) for every β ∈ Q, such that for every r ∈ Z, we have an isomor-phism of filtered D-modules

M (hβ) −→M (hβ−r), ghβ 7−→ (ghr)hβ−r.

For example, we have have an isomorphism of filtered D-modules M (h0) ' OX(∗Z).

Remark 2.14. — Suppose that h, h ∈ OX(X) are nonzero, and α, α ∈ Q>0 are suchthat we have the equality of Q-divisors

α · div(h) = α · div(h).

Let ` be a positive integer such that `α, `α ∈ Z. In this case there is g ∈ O∗X(X)

such that h`α = gh`α. Suppose now that there exists G ∈ OX(X) such that G` = g.

(For example, this holds after pulling-back to the étale cover SpecOX [z]/(z` − g).)In this case we have an isomorphism of filtered DX -modules

Φ: M (h−α) −→M (h−α

)

given byΦ(wh−α) = wG−1h

−α.

Indeed, this follows from the definition of the filtrations and the isomorphism ofschemes over U

ϕ : SpecOU [y]/(y` − h−`α) −→ SpecOU [y]/(y` − h−`α)

that pulls-back y to G−1y.

Remark 2.15. — It is clear that the filtration on M (h−α) is compatible with restric-tion to open subsets. More generally, it is compatible with smooth pullback, as follows.Suppose that h ∈ OX(X) is nonzero and α ∈ Q. If ϕ : Y → X is a smooth morphismand g = h ϕ, then there is an isomorphism of DY -modules

M (g−α) ' ϕ∗M (h−α),

such that for every k we have

FkM (g−α) ' ϕ∗FkM (h−α).

Indeed, choose ` > 2 such that `α ∈ Z and consider the Cartesian diagram

VYpY

//

ψ

UYjY

//

Y

ϕ

Vp

// Uj

// X,

J.É.P. — M., 2019, tome 6


where j and p are as in Lemma 2.6 and jY and pY are the corresponding morphismsfor Y and g. Note that we have a base-change theorem that gives

(2.7) ϕ!j+p+QHV [n] ' (jY )+(pY )+ψ

!QHV [n]

(see [Sai90, (4.4.3)]). Moreover, since ϕ is smooth, if d = dim(Y ) − dim(X), thenfor every filtered D-module (M , F ) underlying a mixed Hodge module M , thefiltered D-module underlying ϕ!M is (ϕ∗M , F )[d], where Fk(ϕ∗M ) = ϕ∗(FkM )

(see [Sai88, 3.5]). This also applies to ψ; in particular, we have ψ!QHV [n] ' QH

VY[n+2d].

By decomposing both sides of (2.7) with respect to the µ`-action, we obtain ourassertion.

3. The case of smooth divisors. — Our goal now is to describe the filtrations intro-duced in the previous section when Z is a smooth divisor. We will then use this todefine Hodge ideals for arbitrary Q-divisors. The key result in the smooth case is thefollowing:

Lemma 3.1. — Letψ : Y = SpecC[t] −→ X = SpecC[x]

be the map given by ψ∗(x) = t`. If Z is the divisor on Y defined by t, then we havean isomorphism of filtered DX-modules

ψ+OY (∗Z) '`−1⊕j=0

Mj ,

where Mj ' DX/DX(∂xx − j/`) and FkMj is generated over OX by the classes of1, ∂x, . . . , ∂

kx . Moreover, if we consider on Y the µ`-action such that every λ ∈ µ`

maps t to λt, then Mj is the component of ψ+OY (∗Z) on which every λ ∈ µ` acts bymultiplication with λj.

Proof. — As usual, it is easier to do the computation for the filtered right D-moduleωY (∗Z) corresponding to OY (∗Z). Note that this is filtered quasi-isomorphic to thecomplex

A• : 0 −→ DYw−−→ ωY (Z)⊗OY DY −→ 0,

placed in degrees −1 and 0, where w(1) = (dt/t) ⊗ t∂t; see e.g. [MP16, Prop. 3.1].Since ψ is finite, the functor ψ∗ is exact on quasi-coherent OY -modules, henceψ+ωY (∗Z) is computed by the 0-th cohomology of the complex

B• = ψ∗(A• ⊗DY DY→X),

with the obvious induced filtration. The definition of w immediately implies thatw ⊗ 1DY→X is injective. Note that dt/t = (1/`)dx/x and t∂t = `x∂x.

In order to describe the complex B•, note that any element of B−1 can be uniquelywritten as

∑`−1j=0 t

jPj , with Pj ∈ DX . Similarly, any element in B0 can be uniquely

J.É.P. — M., 2019, tome 6


written as∑`−1j=0 t

j(dx/x)Qj , with Qj ∈ DX . Moreover, if τ is the differential in B•,then

τ

(`−1∑j=0

tjPj

)=

`−1∑j=0

tjdx

x(x∂x + j/`)Pj ,

where we use the fact that t∂ttj = tjt∂t + jtj . In other words, we have have aneigenspace decomposition

B• '`−1⊕j=0

B•j ,

where B•j is identified with the complex

0 −→ DX −→ DX −→ 0,

with the differential mapping P to (x∂x + j/`)P . It follows that B• is filtered quasi-isomorphic to

`−1⊕j=0

DX/(x∂x + j/`)DX ,

where the filtration on the j-th component is such that

Fk−1 (DX/(x∂x + j/`)DX)

is the OX -submodule generated by the classes of 1, ∂x, . . . , ∂kx . Moreover, every λ ∈ µ`

acts on the jth factor in the above decomposition by multiplication with λj .The assertion in the lemma now follows immediately from the explicit description

of the equivalence between the categories of left and right D-modules on X = A1.Indeed, recall that if τ is the C-linear endomorphism of the Weyl algebra Γ(A1,DA1)

such that τ(PQ) = τ(Q) · τ(P ) for all P and Q, and such that τ(t) = t and τ(∂t) =

−∂t, then the left D-module N corresponding to a right D-module M is isomorphicto M itself, with scalar multiplication given via the map τ . Moreover, for filteredD-modules, via this isomorphism FkN corresponds to Fk−1M . In particular, we seethat if M = DX/P ·DX , then N ' DX/DX · τ(P ), and we obtain the statement.

In what follows, we denote by dαe the smallest integer that is > α. For a Q-divisorD =

∑ri=1 aiDi, we put dDe =

∑ri=1daieDi.

Corollary 3.2. — If h ∈ OX(X) is nonzero and such that the support Z of div(h) issmooth (possibly disconnected), then for every α ∈ Q>0 the filtration on M (h−α) isgiven by

FkM (h−α) = OX((k + 1)Z − dDe

)h−α if k > 0,

where D = α · div(h), and FkM (h−α) = 0 if k < 0.

Proof. — We first reduce to the case when Z = div(h). We can check the assertion inthe proposition locally, hence we may assume that Z = div(g), for some g ∈ OX(X),and h = ugm, for some u ∈ O∗X(X). Furthermore, by Remark 2.15, it is enough toprove the assertion after passing to a surjective étale cover, hence we may assumethat u = vm for some v ∈ O∗X(X). After replacing g by vg, we may thus assume

J.É.P. — M., 2019, tome 6


that h = gm. In this case we have an isomorphism of filtered D-modules M (h−α) 'M (g−mα), hence we may and will assume that div(h) = Z.

We consider the smallest positive integer ` such that m := `α ∈ Z. If ` = 1, thenthe assertion follows from the formula for the filtration on OX(∗Z) when Z is smooth;see [MP16, Prop. 8.2]. Therefore from now on we assume ` > 1.

The morphism h : X → A1 is smooth over some open neighborhood of 0. UsingRemark 2.15, we see that in order to prove the corollary, we may assume that X = A1

and h = x, the standard coordinate on A1. Consider the Cartesian diagram

Vj0//

p

W

g

Uj// X,

where

j0 : V = SpecC[x, x−1, y]/(y`−x−m) −→W = SpecC[x, z]/(z`−xm), j∗0 (z) = y−1.

Let ϕ : W = SpecC[t]→W be the normalization, given by

ϕ∗(x) = t` and ϕ∗(z) = tm.

(Here we use that ` and m are relatively prime.) Note that ϕ is an isomorphismover V , hence we have an open embedding ι : V → W , with complement the smoothdivisor T defined by t (in fact, if a and b are integers such that am + b` = 1, thenι∗(t) = y−axb). We thus have

j+p+OV ' ψ+ι+OV ' ψ+OW

(∗T ),

where ψ = gϕ. We apply Lemma 3.1 for ψ. Note that ϕ is a µ`-equivariant morphismif we let each λ ∈ µ` act on t by multiplication with λa. By considering the behaviorwith respect to the µ`-action, we see that in the decomposition given by the lemma,we have Mj 'M (x−α) if and only if ja ≡ −1 (mod `), that is, j ≡ −m (mod `).

Suppose first that α < 1, in which case the condition for j is that j = `−m. As areality check, note that we indeed have an isomorphism

DX/DX(∂xx− (`−m)/`) 'M (x−α)

that maps the class of 1 to x−α. The formula for the filtration on M (hα) now followsfrom Lemma 3.1. When α > 1, we putm = dαe−1, and use the fact from Remark 2.4,namely that we have an isomorphism of filtered modules

M (x−α) −→M (x−α+m), gx−α 7−→ (gx−m)x−α+m,

to reduce the assertion to the case α ∈ (0, 1). This completes the proof of the corollary.

J.É.P. — M., 2019, tome 6


4. Definition of Hodge ideals for Q-divisors. — In general, we obtain an upperbound for the terms in the filtration on M (h−α) by restricting to the open subsetwhere the support of div(h) is smooth, as follows.

Proposition 4.1. — Given a nonzero h ∈ OX(X) and a positive rational number α,for every k > 0 we have

FkM (h−α) ⊆ OX((k + 1)Z − dDe

)h−α,

where D = α · div(h) and Z = Supp(D), while FkM (h−α) = 0 for k < 0.

Proof. — Let ι : X0 → X be an open immersion such that the codimension of itsimage in X is > 2 and Z|X0

is smooth (though possibly disconnected). Note thatour constructions are compatible with restrictions to open subsets. Moreover, sinceM (h−α) is clearly torsion-free, it follows that Fk := FkM (h−α) is torsion free, hencethe canonical map Fk → ι∗

(Fk|X0

)is injective. Therefore it is enough to prove the

assertion on X0, hence we may assume that Z is smooth. However, in this case theassertion follows from Corollary 3.2.

We can now define the Hodge ideals for Q-divisors. Let X be a smooth com-plex algebraic variety and Z a reduced effective divisor on X. Given an effectiveQ-divisor D with Supp(D) = Z, we define coherent ideals sheaves Ik(D) in OX asfollows. Suppose first that there is a nonzero h ∈ OX(X), with H = div(h), and apositive rational number α such that D = αH. It turns out to be more convenient towork with the DX -module M (hβ), where β = 1 − α. Recall that we have a filteredisomorphism

M (h−α) −→M (hβ), wh−α 7−→ (wh−1)hβ ,

and therefore, if k > 0, it follows from Proposition 4.1 that there is a unique coherentideal Ik(D) such that

FkM (hβ) = Ik(D)⊗OX OX(kZ +H

)hβ

(note that we always have dDe > Z). The definition is independent of the choiceof α and h: indeed, using Remark 2.15, it is enough to check this after the pullbackby a suitable étale surjective map, hence we deduce the independence assertion usingRemark 2.14. This implies that the general case of the definition follows by covering Xwith suitable affine open subsets such that D can be written as above in each of them.Note that when D = Z we have β = 0, and so the ideals Ik(D) are the Hodge idealsstudied in [MP16].

Remark 4.2. — From the definition and the filtration property, it follows that wealways have the inclusion

OX(−Z) · Ik−1(D) ⊆ Ik(D) for k > 1.

We note that for the reduced divisor Z, we have the more subtle inclusions

Ik(Z) ⊆ Ik−1(Z) for k > 1

J.É.P. — M., 2019, tome 6


(see [MP16, Prop. 13.1]). We do not know however whether this holds for arbitraryQ-divisors D, and in fact we suspect that this is not the case. (Note that it doeshold when D has simple normal crossings support by Proposition 7.1. It is also shownto hold when D has an isolated weighted homogeneous singularity in the upcoming[Zha18].) However, when D = αZ these inclusions do hold modulo the ideal OX(−Z),see [MP18a, Cor. B]. More precisely, we have

Ik(αZ) + OX(−Z) ⊆ Ik−1(αZ) + OX(−Z) for k > 1.

This implies in particular that if Ik(αZ) = OX for some k > 1, then Ik−1(αZ) = OX .

Remark 4.3. — According to Proposition 4.1, we also have ideals I ′k(D) given by

FkM (h−α) = I ′k(D)⊗OX OX((k + 1)Z − dDe

)h−α,

which are related to Ik(D) by the formula

Ik(D) = I ′k(D)⊗OX OX(Z − dDe).

The following periodicity property often allows us to reduce our study to the casedDe = Z.

Lemma 4.4. — If D′ is an integral divisor with Supp(D′) ⊆ Supp(D), then

Ik(D +D′) = Ik(D)⊗OX OX(−D′).

In particularIk(D) = Ik(B)⊗ OX(Z − dDe),

with B = D + Z − dDe satisfying dBe = Z.

Proof. — Using the notation in Remark 4.3, the equivalent statement

I ′k(D +D′) = I ′k(D)

follows from the definition and Remark 2.13.

Remark 4.5. — Note that Ik(D) ⊆ OX(Z − dDe) for all k, and so if dDe 6= Z,then one can never have Ik(D) = OX . It is however still interesting to ask whetherIk(B) = OX .

5. A global setting for the study of Hodge ideals. — We now consider a settingin which we can define global filtered DX -modules that are locally isomorphic to the(M (h−α), F

)discussed in the previous sections.

Let X be a smooth variety and D = (1/`)H a Q-divisor, where H is an integraldivisor and ` is a positive integer. The extra assumption we make here is that thereis a line bundle M such that

OX(H) 'M⊗`.

We denote by U the complement of Z = Supp(H) and by j the inclusion U → X.

J.É.P. — M., 2019, tome 6


Let s ∈ Γ(X,M⊗`) be a section whose zero locus is H. Since s does not vanishon U , we may consider the section s−1 ∈ Γ

(U, (M−1)⊗`

). Let p : V → U be the étale

cyclic cover corresponding to s−1, hence

V ' Spec(OU ⊕M ⊕ · · · ⊕M⊗(`−1)

).

We consider the filtered DX -module M = j+p+OV , that underlies a mixed Hodgemodule. The µ`-action on V , where λ ∈ µ` acts on M⊗i by multiplication with λ−i,induces an eigenspace decomposition

M =`−1⊕i=0

Mi,

where λ ∈ µ` acts on Mi by multiplication with λ−i. We consider on each Mi theinduced filtration.

Note that if X0 is an open subset of X such that we have a trivialization M |X0 'OX0 , and if via the corresponding trivialization of M⊗`|X0 , the restriction s|X0 cor-responds to h0 ∈ OX(X0), then we have isomorphisms of filtered DX0

-modules

Mi 'M (h−i/`0 ) for i such that 0 6 i 6 `− 1.

We also see that the filtration on M is the direct sum filtration, since this holdslocally. Moreover, we have isomorphisms of OX0

-modules

Mi|X0' OX(∗Z)|X0

,

which glue to isomorphisms of OX -modules

Mi 'M⊗i ⊗OX OX(∗Z) = j∗j∗M⊗i.

Via these isomorphisms, it follows from the definition of Hodge ideals (see also Re-mark 4.3) that we have

FkMi 'M⊗i ⊗OX I′k (i/` ·H)⊗OX OX ((k + 1)Z − di/` ·He)

'M⊗i(−H)⊗OX Ik (i/` ·H)⊗OX OX (kZ +H) .

6. A complex associated to simple normal crossing divisors. — We now discuss acomplex that, as we will see later, gives a filtered resolution of Mr(h

−α) by filteredinduced DX -modules in the case when h defines a simple normal crossing divisor.

Let X be a smooth, n-dimensional, complex variety, h ∈ OX(X) nonzero,and α a nonzero rational number (we allow α to be either positive or negative).Let D = α · div(h). We denote by Z the support of D, and assume that it has simplenormal crossings.

Associated to Z we have the following complex of right DX -modules:

C• : 0 −→ DX −→ Ω1X(logZ)⊗OX DX −→ · · · −→ ΩnX(logZ)⊗OX DX −→ 0,

placed in degrees −n, . . . , 0. We denote by Di : Ci → Ci+1 its differentials. If

x1, . . . , xn are local coordinates on X, then

Di(η ⊗ P ) = dη ⊗ P +

n∑i=1

(dxi ∧ η)⊗ ∂xiP.

J.É.P. — M., 2019, tome 6


In fact C• is a filtered complex, where

Fp−nCi = Ωi+nX (logZ)⊗OX Fp+iDX .

This filtered complex is quasi-isomorphic to the filtered right DX -module ωX(∗Z)

corresponding to the filtered left DX -module OX(∗Z) (see [MP16, Prop. 3.1], and[Sai90, Prop. 3.11(ii)] for a more general statement).

Given h and α as above, we also consider the filtered complex C•h−α consisting ofthe same sheaves, but with differential Cih−α → Ci+1

h−α given by

Di −((α · d log(h) ∧ •)⊗ 1DX

).(1)

It is easy to see that this is indeed a filtered complex.Suppose now that we also have an effective divisor T supported on Z. It is not

hard to check that the formula for the map

Cih−α −→ Ci+1h−α

induces also a map

Cih−α(−T ) := OX(−T )⊗OX Ωi+nX (logZ)⊗OX DX

−→ Ci+1h−α(−T ) := OX(−T )⊗OX Ωi+1+n

X (logZ)⊗OX DX .

This is due to the fact that if locally T = div(u) and η is a local section of Ωi+nX (logZ),then we can write d(uη) = ud(η)+u·d log(u)∧η. We thus obtain a filtered subcomplexC•h−α(−T ) of C•h−α . We emphasize that this is not obtained by tensoring C•h−α withOX(−T ).

Proposition 6.1. — If no coefficient of D−T lies in Z<0, then the complex C•h−α(−T )

is filtered quasi-isomorphic to(h−αωX(∗Z), G•

), where

Gk−nh−αωX(∗Z) = 0 if k < 0,

G−nh−αωX(∗Z) = h−αωX(Z − T )

Gk−nh−αωX(∗Z) = G−nh

−αωX(∗Z) · FkDX if k > 0.and

Proof. — It is immediate to check that the differential induced on grFp C•h−α(−T )

does become equal to the differential Di twisted with the identity on OX(−T ), andtherefore for every p we have

grFp C•h−α(−T ) = OX(−T )⊗OX grFp C

•.

In particular, we have

HiFpC•h−α(−T ) = 0 for every p ∈ Z and i ∈ Zr 0,

by the result in [MP16] quoted above. Consider now the morphism of right DX -mod-ules

ϕ : C0h−α(−T ) = ωX(Z − T )⊗OX DX −→ h−αωX(∗Z)

(1)In related settings, for instance involving the de Rham complex of M (h−α), this type ofcomplex can already be found in the literature; see for instance [Bjö93, §6.3.11].

J.É.P. — M., 2019, tome 6


given byϕ(w ⊗ η ⊗Q) = (h−αwη)Q.

We first check that this morphism is surjective. We do this locally, hence we mayassume that we have a system of coordinates x1, . . . , xn on X such that OX(−Z) isgenerated by x1 · · ·xr and OX(−T ) by xβ1

1 · · ·xβrr . We also write h = uxa11 · · ·xarr ,where u is an everywhere nonvanishing function, and define αi = αai and γi = αi−βifor all i. Note for later use that

α · d log(h) =α

udu+

r∑i=1

αidxixi.

The surjectivity of ϕ follows from the fact that

(6.1) Im(ϕ) = (h−αxβ1

1 · · ·xβrr η) ·DX = h−αωX(∗Z),

whereη = d log(x1) ∧ · · · ∧ d log(xr) ∧ dxr+1 ∧ · · · ∧ dxn,

and the second equality in (6.1) is a consequence of the fact that −γi − 1 6∈ Z>0 forall i, by assumption.

In order to complete the proof of the proposition it is enough to show that, forevery k > 0, the following sequence is exact:

OX(−T )⊗ Ωn−1X (logZ)⊗ Fk−1DXψk−−−→ OX(−T )⊗ ωX(Z)⊗ FkDX

ϕk−−−→ Gk−nh−αωX(∗Z) −→ 0,

where ϕk is the restriction of ϕ to the (k − n)-th level of the filtration and ψk is therestriction of the differential of C•h−α(−T ). The surjectivity of ϕk is an immediateconsequence of the surjectivity of ϕ and the definition of the filtration on hαωX(∗Z).

Keeping the above notation for the local coordinates on X, it follows from thedefinition of ψk that

Im(ψk) =

r∏j=1

xβjj ⊗ η ⊗

( r∑i=1

(xi∂i − γi −

∂u

∂xi· αxiu

)· Fk−1DX

+

n∑i=r+1

(∂i −

∂u

∂xi· αu

)· Fk−1DX

)and it is straightforward to see that this is contained in Ker(ϕk). We now proveby induction on k that if ϕk(xβ1

1 · · ·xβrr ⊗ η ⊗ P ) = 0 for some P ∈ FkDX , thenxβ1

1 · · ·xβrr ⊗ η ⊗ P ∈ Im(ψk). Note that the case k = 0 is trivial. Let us writeP =

∑u,v cu,v∂

uxv, where u and v vary over Zn>0. After subtracting suitable termsfrom P , we may assume that whenever cu,v 6= 0, we have ui = 0 for i > r. Furthermore,note that if ui, vi > 0 for some i 6 r, then we can write

∂uxv =(xi∂i − γi −

∂u

∂xi· αxiu

)A+B,

J.É.P. — M., 2019, tome 6


with both A and B of order 6 k − 1. Therefore we may also assume that whenevercu,v 6= 0 and |u| :=

∑i ui = k, we have

(6.2) uivi = 0 for i such that 1 6 i 6 n.

Since

(h−αxβ1

1 · · ·xβrr η)∂uxv = (non-zero constant(2)) · (h−αxβ1

1 · · ·xβrr η)xv−u,

and since (6.2) implies that for every (u, v) and (u′, v′) with |u| = k and cu,v, cu′,v′ 6= 0

we have xv−u 6= xv′−u′ , we conclude that in fact P ∈ Fk−1DX , hence we are done by

induction.

7. The Hodge ideals of simple normal crossing divisors. — In this section we showthat the Hodge ideals of divisors with simple normal crossing support essentiallydepend only on the support of the divisor, and therefore can be computed as in[MP16, §8].

Proposition 7.1. — Let X be a smooth variety, and D an effective divisor on X withsimple normal crossing support Z. Then for all k we have

Ik(D) = Ik(Z)⊗OX OX(Z − dDe).

Proof. — Equivalently, we need to show that I ′k(D) = Ik(Z) for all k. The assertionis local, hence we may assume that we have coordinates x1, . . . , xn on X such thatZ = H1 + · · ·+Hr, where Hi is defined by xi = 0. The morphism X → Cr given by(x1, . . . , xr) is smooth, hence using Remark 2.15 we see that it is enough to prove theproposition when X = SpecC[x1, . . . , xn] and D =

∑ni=1 αiHi, where Hi = div(xi)

and αi > 0. Let ` be the smallest positive integer such that all ai := `αi are integers.The assertion to be proved is trivial when ` = 1, hence from now on we assume ` > 2.Consider the Cartesian diagram

Vj0//

p

W

g

Uj// X,

where

j0 : V = SpecC[x±11 , . . . , x±1n , y]/(y` − x−a11 · · ·x−ann )

−→W = SpecC[x1, . . . , xn, z]/(z` − xa11 · · ·xann ),

with j∗0 (z) = y−1. We will make use of some standard facts about cyclic covers withrespect to simple normal crossing divisors, exploiting the toric variety structure onthe normalization of W . For basic facts regarding toric varieties, we refer to [Ful93].

Let N be the lattice Zn and M its dual. We also consider the lattice

N ′ = (v1, . . . , vn+1) ∈ Zn+1 | a1v1 + · · ·+ anvn = `vn+1

(2)Here we use again the fact that −γi − 1 6∈ Z>0 for all i.

J.É.P. — M., 2019, tome 6


and its dualM ′ = Zn+1/Z · (a1, . . . , an,−`).

Note that we have an injective lattice map N ′ → N , with finite cokernel, inducedby the projection onto the first n components, and the dual map M → M ′ is againinjective, with finite cokernel. In fact, we have an isomorphism M ′/M ' Z/`Z thatmaps the class of (u1, . . . , un+1) ∈M ′ to the class of un+1 in Z/`Z.

We thus have an isomorphism N ′R ' NR = Rn. The strongly convex cone σ = Rn>0

in NR = Rn gives the toric variety X = Cn. As a cone in N ′R, σ gives an affine toricvariety W , and the lattice map N ′ → N corresponds to a toric map ψ : W → X.Note that we have a morphism of O(X)-algebras O(W )→ O(W ) that maps xi to theelement of C[σ∨ ∩M ′] corresponding to the class of the i-th element of the standardbasis of Zn, and z to the class of (0, . . . , 0, 1). It is easy to check that if we denoteby bγc the largest integer 6 γ, then

(7.1) O(W ) =⊕

06j6`−1O(X)x

−bjα1c1 · · ·x−bjαncn zj ,

and consequently to deduce that O(W ) is integral over O(W ). As the coordinate ringof a toric variety, O(W ) is normal, hence it is the integral closure of O(W ) in its fieldof fractions. Moreover, since W is a toric variety, we may choose a toric resolution ofsingularities Y → W , and let f : Y → X be the composition. Since the map Y → W

is an isomorphism over the complement of g−1(∑Hi), it follows that there is an open

embedding ι : V → Y such that f ι = j p. The support EY of Y r ι(V ) is the sumof all prime toric divisors on Y .

The action of the torus TM ′ = SpecC[M ′] on W induces an action of the finitegroup SpecC[M ′/M ] ' SpecC[Z/`Z] = µ` on W . This is the action induced onthe normalization W by the µ`-action on W that we discussed in Section 2. In par-ticular, the toric resolution Y → W is automatically equivariant. Note that in thedecomposition (7.1), an element λ ∈ µ` acts on the summand corresponding to j bymultiplication with λj .

The equality f ι = j p implies that we have an isomorphism of filtered DX -mod-ules

j+p+OV ' f+ι+OV = f+OY (∗EY ).

As usual, in order to compute the push-forward of OY (∗EY ), it is more convenient towork with right D-modules. Recall that there is a complex of right DY -modules

A• = A•Y : 0 −→ DY −→ Ω1Y (logEY )⊗OY DY −→ · · · −→ ωY (EY )⊗OY DY −→ 0

located in degrees −n, . . . , 0, that is filtered quasi-isomorphic to ωY (∗EY ); see thebeginning of Section 6. Since Y is a toric variety, we have a canonical isomorphismΩ1Y (logEY ) 'M ′ ⊗Z OY (see [Ful93, §4.3]). We will also consider the corresponding

complex on X:

A•X : 0 −→ DX −→M ⊗Z DX −→ · · · −→ ∧nM ⊗Z DX −→ 0.

J.É.P. — M., 2019, tome 6


It follows from the definition that, forgetting about the filtration, we have

f+ωY (∗EY ) = Rf∗(A• ⊗DY DY→X).

Note that DY→X = f∗DX as OY -modules, hence the projection formula implies

Rif∗(Ap−n ⊗DY DY→X) ' ∧pM ′ ⊗Z R

if∗OY ⊗OX DX = 0

for i > 0, since f is the composition of a finite map with a toric resolution. Thereforef+ωY (∗EY ) is represented by the complex B•, where

Bp−n = ∧pM ′ ⊗Z ψ∗OW ⊗OX DX .

In order to describe the differential of this complex, it is convenient to use the iso-morphism MQ 'M ′Q and the decomposition (7.1). With a little care, it follows fromthe definitions that if we put

Bp−nj = ∧pMQ ⊗Q OX · x−bjα1c1 · · ·x−bjαncn zj ⊗OX DX ,

then B• decomposes as the direct sum of the subcomplexes B•j , for j such that 0 6

j 6 `− 1. Furthermore, if we identify each Bp−nj in the obvious way with Ap−nX , thenthe differential

δp−nBj: ∧pMQ ⊗Q DX −→ ∧p+1MQ ⊗Q DX

is given byδp−nBj

= δp−nAX+ (wj ∧ −)⊗ IdDX ,

where δAX is the differential on A•X and

wj = (wj,1, . . . , wj,n), with wj,i = jαi − bjαic.

It follows from Proposition 6.1 that we have a morphism

B0j −→Mr(x

wj,11 · · ·xwj,nn )

that induces a quasi-isomorphism

B•j −→Mr(xwj,11 · · ·xwj,nn )

(see also Remark 2.3).We now bring the filtrations into the picture. It follows from Saito’s strictness

results (see the discussion in Section 1; cf. also [MP16, §4, §6]) that

Fkf+ωY (∗EY ) = Im(Rf∗Fk(A• ⊗DY DY→X) −→ Rf∗(A

• ⊗DY DY→X)).

Arguing as above, we deduce that

Fkf+ωY (∗EY ) = Im(f∗Fk(A• ⊗DY DY→X) −→ f∗(A

• ⊗DY DY→X)).

In other words, (f+ωY (∗EY ), F ) is represented by the filtered complex B•, and usingProposition 6.1, we conclude that

f+ωY (∗EY ) '`−1⊕j=0

Mr(xwj,11 · · ·xwj,nn ),

J.É.P. — M., 2019, tome 6


where the filtration on Mr(xwj,11 · · ·xwj,nn ) is given by

F−nMr(xwj,11 · · ·xwj,nn ) = x

wj,11 · · ·xwj,nn ωX(Z)

Fk−nMr(xwj,11 · · ·xwj,nn ) = F−nMr(x

wj,11 · · ·xwj,nn ) · FkDX for k > 1.and

By comparing the µ`-actions, we conclude that the summand Mr(x−α11 · · ·x−αnn )

on which an element λ ∈ µ` acts by multiplication with λ−1 corresponds to j = `− 1.Therefore the filtration on M (x−α1

1 · · ·x−αnn ) is given by

F−nMr(x−α11 · · ·x−αnn ) = (x−α1

1 · · ·x−αnn )xdα1e1 · · ·xdαnen ωX(Z)

Fk−nMr(x−α11 · · ·x−αnn ) = F−nMr(x

−α11 · · ·x−αnn ) · FkDX for k > 1.and

It is now a straightforward computation to see that I ′k(D) is the ideal generated bythe monomials

∏ni=1 x

cii , where 0 6 ci 6 k for all i and

∑i ci = (n − 1)k. This

coincides with Ik(Z) according to [MP16, Prop. 8.2], completing the proof of theproposition.

8. Computation in terms of a log resolution. — We use the results of the previoustwo sections in order to describe Hodge ideals of Q-divisors in terms of log resolutions.Let X be a smooth variety, h ∈ OX(X) a nonzero function, H = div(h), and α ∈ Q>0.We are interested in computing Ik(D), where D = αH. As always, let Z = Supp(D)

and β = 1− α.Let f : Y → X be a log resolution of the pair (X,D) that is an isomorphism over

U = X r Z, and denote g = h f ∈ OY (Y ). We fix a positive integer ` such that`α ∈ Z. As usual, we consider

p : V = SpecOU [y]/(y` − h−`α) −→ U

and the inclusion j : U → X. By assumption, we also have an open immersionι : U → Y such that f ι = j. By considering the decompositions of

j+p+OV ' f+ι+p+OV

into isotypical components, we conclude that we have a filtered isomorphism

(8.1) M (h−α) ' f+M (g−α).

We now denote G = f∗D, and consider on Y the complex introduced in Section 6:

C•g−α(−dGe) : 0 −→ OY (−dGe)⊗OY DY −→ OY (−dGe)⊗OY Ω1Y (logE)⊗OY DY

−→ · · · −→ OY (−dGe)⊗OY ωY (E)⊗OY DY −→ 0,

where E = (f∗D)red. This is placed in degrees −n, . . . , 0, and if x1, . . . , xn are localcoordinates on Y , then its differential is given by

η ⊗Q 7−→ dη ⊗Q+

n∑i=1

(dxi ∧ η)⊗ ∂iQ−(α · d log(g) ∧ η

)⊗Q.

J.É.P. — M., 2019, tome 6


Theorem 8.1. — With the above notation, the following hold:(i) For every p 6= 0 and every k ∈ Z, we have

Rpf∗(C•g−α(−dGe)⊗DY DY→X

)= 0

Rpf∗Fk(C•g−α(−dGe)⊗DY DY→X

)= 0.and

(ii) For every k ∈ Z, the natural inclusion induces an injective map

R0f∗Fk(C•g−α(−dGe)⊗DY DY→X

)−→ R0f∗


).

(iii) We have a canonical isomorphism

R0f∗(C•g−α(−dGe)⊗DY DY→X

)'Mr(h

−α)

that induces for every k ∈ Z an isomorphism

R0f∗Fk−n(C•g−α(−dGe)⊗DY DY→X

)' h−αωX

((k + 1)Z − dDe

)⊗OX I

′k(D)

' hβωX(kZ +H

)⊗OX Ik(D).

Proof. — It follows from Lemma 2.8, and from the definition of its filtration, thatMr(g

−α) is a direct summand of a right Hodge D-module on Y . By Saito’s strictnessof the filtration of (push-forwards of) such D-modules, it follows that for all k, p ∈ Z

the canonical map

Rpf∗Fk(Mr(g

−α)L⊗DY DY→X

)−→ Rpf∗

(Mr(g

−α)L⊗DY DY→X

)is injective, and its image is equal to

FkRpf∗(Mr(g

−α)L⊗DY DY→X

)(see the discussion in Section 1).

On the other hand, note that if write G = α ·div(g) =∑i αiEi, then −dαie+αi 6∈

Z<0 for all i. We may thus apply Proposition 6.1 for the divisor G, with T = dGe.Using Proposition 7.1 as well, we see that C•g−α(−dGe) is filtered quasi-isomorphic toMr(g

−α), hence

Rpf∗(C•g−α(−dGe)⊗DY DY→X

)' Rpf∗

(Mr(g

−α)L⊗DY DY→X

)Rpf∗Fk


)' Rpf∗Fk

(Mr(g

−α)L⊗DY DY→X

).and

Finally, by the definition of push-forward for right D-modules we have

Rpf∗(Mr(g

−α)L⊗DY DY→X

)' Hpf+Mr(g

−α),

and by (8.1) this is 0 if p 6= 0, and is canonically isomorphic to Mr(h−α) if p = 0.

The assertions in the proposition follow by combining all these facts.

Remark 8.2 (Local vanishing). — The statement in Theorem 8.1 (i) is a generalizationof the Local Vanishing theorem for multiplier ideals [Laz04, Th. 9.4.1], in view of thecalculation in Proposition 9.1 below.

J.É.P. — M., 2019, tome 6


As a consequence of the vanishing statements in Theorem 8.1 (i), provided bystrictness, we deduce the following local Nakano-type vanishing result, first obtainedby Saito [Sai07, Cor. 3] when D is reduced; cf. Corollary C in the introduction andthe discussion following it.

Corollary 8.3. — Let D be an effective Q-divisor on the smooth variety X andf : Y → X a log resolution of (X,D) that is an isomorphism over X r Supp(D). IfE = (f∗D)red, then

Rqf∗(OY (−df∗De)⊗OY ΩpY (logE)

)= 0 for p+ q > n = dim(X).

Proof. — We argue by descending induction on p, the case p > n being trivial. Sup-pose now that p 6 n and q > n − p. After possibly replacing X by suitable opensubsets, we may assume that D = α · div(h). We may thus apply Theorem 8.1 todeduce that if

C• = F−n(C•g−α(−df∗De)⊗DY DY→X

)[p− n],

then

(8.2) Rjf∗C• = 0 for j > n− p.

Note that by definition, we have

Ci = OY (−df∗De)⊗OY Ωp+iY (logE)⊗OY f∗FiDX for i such that 0 6 i 6 n− p.

Consider the spectral sequence

Ei,j1 = Rjf∗Ci ⇒ Ri+jf∗C

•.

It follows from (8.2) that E0,q∞ = 0. Now by the projection formula we have

(8.3) Ei,j1 = Rjf∗(OY (−df∗De ⊗OY Ωp+iY (logE)

)⊗OX FiDX .

In particular, it follows from the inductive hypothesis that for every r > 1 we haveEr,q−r+1

1 = 0, hence Er,q−r+1r = 0 as well. On the other hand, we clearly have

E−r,q+r−1r = 0, since this is a first-quadrant spectral sequence. We thus conclude that

E0,qr = E0,q

r+1 for all r > 1,

hence E0,q1 = E0,q

∞ = 0. Using (8.3) again, we conclude that

Rqf∗(OY (−df∗De)⊗OY ΩpY (logE)

)= 0.

9. The ideal I0(D) and log canonical pairs. — We now use Theorem 8.1 in orderto relate I0(D) to multiplier ideals. Recall that for a Q-divisor B, one denotes byI (B) the associated multiplier ideal; see [Laz04, Ch. 9] for the definition and basicproperties.

Proposition 9.1. — If f : Y → X is a log resolution of (X,D) that is an isomorphismover X rD, and E = (f∗D)red, then

I0(D) ' f∗OY(KY/X + E − df∗De

)= I

((1− ε)D

)for ε such that 0 < ε 1.

J.É.P. — M., 2019, tome 6


Proof. — The first equality follows from Theorem 8.1, together with the fact that theterm F−nC

•g−α(−df∗De) consists of

ωY (E − df∗De)

placed in degree 0. The second equality then follows from the definition of multiplierideals and the fact that if A is an effective divisor with support E, then

E − dAe = −b(1− ε)Ac for ε such that 0 < ε 1.

As in [MP16] in the case of reduced divisors, we obtain therefore that for everyQ-divisor D we have that I0(D) = OX if and only if the pair (X,D) is log canonical,which leads to the following:

Definition 9.2. — The pair (X,D) is k-log canonical if

I0(D) = · · · = Ik(D) = OX .(3)

Note however that by Remark 4.3, the triviality of any Ik(D) is possible only ifdDe = Z; in general it is more suitable to focus on the triviality of the ideals I ′k(D).We therefore introduce also:

Definition 9.3. — The pair (X,D) is reduced k-log canonical if

I ′0(D) = · · · = I ′k(D) = OX ,

or equivalentlyI0(D) = · · · = Ik(D) = OX(Z − dDe).

Example 9.4. — Let Z have an ordinary singularity, i.e., an isolated singular pointwhose projectivized tangent cone is smooth, of multiplicity m. If D = αZ with0 < α 6 1, then

(X,D) is k-log canonical ⇐⇒ k 6[ nm− α

].

See Corollary 11.8 and Remark 11.9.

C. Local study and global vanishing theorem

10. Generation level of the Hodge filtration, and examples. — As above, we con-sider a divisor D = αH, with H = div(h) for some nonzero h ∈ OX(X) and α ∈ Q>0.We denote by Z the support of D, and β = 1 − α. By construction, the filtrationon M (hβ) is compatible with the order filtration on DX . This means that for everyk, ` > 0 we have

(10.1) F`DX ·(Ik(D)⊗ OX(kZ +H)hβ

)⊆ Ik+`(D)⊗ OX((k + `)Z +H)hβ ,

or equivalently for every k > 0 we have

(10.2) F1DX ·(Ik(D)⊗ OX(kZ +H)hβ

)⊆ Ik+1(D)⊗ OX((k + 1)Z +H)hβ .

(3)We note that by the results in [MP18a, §5], at least in the case of divisors of the form D = αZ,with α ∈ Q>0, this condition is equivalent simply to Ik(D) = OX (cf. Remark 4.2).

J.É.P. — M., 2019, tome 6


By working locally, we may assume that we also have an equation g for Z. With thisnotation, condition (10.2) is equivalent to the following two conditions:

(10.3) g · Ik(D) ⊆ Ik+1(D)

and for every derivation Q of OX and every w ∈ Ik(D), we have

(10.4) g ·Q(w)− kw ·Q(g)− αgw · Q(h)

h∈ Ik+1(D).

We now turn to the problem of describing the generation level of the filtration onM (hβ). Recall that one says that the filtration is generated at level k if

F`DX · FkM (hβ) = Fk+`M (hβ) for all ` > 0,

or in other words if equality is satisfied in (10.1). This is of course equivalent to having

F1DX · FpM (hβ) = Fp+1M (hβ) for all p > k.

Suppose now that we are in the setting of Theorem 8.1.

Theorem 10.1. — The filtration on M (hβ) is generated at level k if and only if

Rqf∗(Ωn−qY (logE)⊗OY OY (−df∗De)

)= 0 for q > k.

In particular, the filtration is always generated at level n− 1.

Proof. — The proof follows almost verbatim that of [MP16, Th. 17.1]. It is moreconvenient to work equivalently with M (h−α), and in fact with the associated rightDX -module Mr(h

−α). It is enough to show that

(10.5) Fk−nMr(h−α) · F1DX = Fk−n+1Mr(h

−α)

if and only ifRk+1

(f∗Ω

n−k−1Y (logE)⊗OY OY (−df∗De)

)= 0.

The inclusion “⊆” in (10.5) always holds of course by the definition of a filtration,hence the issue is the reverse inclusion.

With the notation in Section 6, for every p let

C•p := Fp(C•g−α(−df∗De)⊗DY DY→X

),

where g = h f . Consider the morphism of complexes

Φk : C•k−n ⊗f−1OX f−1F1DX −→ C•k+1−n

induced by right multiplication, and let T • = Ker(Φk). Using Theorem 8.1, we seethat (10.5) holds if and only if the morphism

(10.6) R0f∗C•k−n ⊗OX F1DX −→ R0f∗C

•k+1−n

induced by Φk is surjective.For every m > 0, let Rm be the kernel of the morphism induced by right multipli-

cationFmDX ⊗OX F1DX −→ Fm+1DX .

J.É.P. — M., 2019, tome 6


Note that this is a surjective morphism of locally free OX -modules, hence Rm is alocally free OX -module and for every p we have

T p = OY (−df∗De)⊗OY Ωn+pY (logE)⊗f−1OX f−1Rk+p.

Consider the first-quadrant hypercohomology spectral sequence

Ep,q1 = Rqf∗Tp−n =⇒ Rp+q−nf∗T

•.

The projection formula gives

Rqf∗Tp−n ' Rqf∗

(OY (−df∗De)⊗OY ΩpY (logE)

)⊗OX Rk+p−n,

and this vanishes for p + q > n by Corollary 8.3. We thus deduce from the spectralsequence that Rjf∗T • = 0 for all j > 0.

We first consider the case when k > n and show that (10.5) always holds. Indeed,in this case Φk is surjective. It follows from the projection formula and the long exactsequence in cohomology that we have an exact sequence

R0f∗C•k−n ⊗OX F1DX −→ R0f∗C

•k+1−n −→ R1f∗T

•.

We have seen that R1f∗T• = 0, hence the morphism in (10.6) is surjective.

Suppose now that 0 6 k < n. Let B• → C•k+1−n be the subcomplex given byBp = Cpk+1−n for all p 6= −k − 1 and B−k−1 = 0. Note that we have a short exactsequence of complexes

(10.7) 0 −→ B• −→ C•k+1−n −→ C−k−1k+1−n[k + 1] −→ 0.

It is clear that Φk factors as

C•k−n ⊗f−1OX f−1F1DX

Φ′k−−−→ B• −→ C•k+1−n.

Moreover, Φ′k is surjective and Ker(Φ′k) = T •. As before, since R1f∗T• = 0, we

conclude that morphism induced by Φ′k:

R0f∗C•k−n ⊗OX F1DX −→ R0f∗B

•

is surjective. This implies that (10.6) is surjective if and only if the morphism

(10.8) R0f∗B• −→ R0f∗C

•k+1−n

is surjective. The exact sequence (10.7) induces an exact sequence

R0f∗B• −→ R0f∗C

•k+1−n −→ Rk+1f∗C

−k−1k+1−n −→ R1f∗B

•.

We have seen that R2f∗T• = 0, and we also have

R1f∗(C•k−n ⊗f−1OX f

−1F1DX

)= 0.

This follows as above, using the projection formula, the hypercohomology spectralsequence, and Corollary 8.3. We deduce from the long exact sequence associated to

0 −→ T • −→ C•k−n ⊗f−1OX f−1F1DX −→ B• −→ 0

J.É.P. — M., 2019, tome 6


that R1f∗B• = 0. Putting all of this together, we conclude that (10.6) is surjective if

and only if Rk+1f∗C−k−1k+1−n = 0. Since by definition we have

Rk+1f∗C−k−1k+1−n = Rk+1f∗

(OY (−df∗De)⊗OY Ωn−k−1Y (logE)

),

this completes the proof of the first assertion in the proposition. The second assertionfollows from the first, since all fibers of f have dimension < n.

Example 10.2 (Nodal curves). — If X is a smooth surface and Z is a reduced curveon X, defined by h ∈ O(X), such that Z has a node at x ∈ X and no other singular-ities, then the filtration on M (hβ) is generated at level 0. Indeed, let f : Y → X bethe blow-up of X at x, with exceptional divisor F . This is a log resolution of (X,Z),hence our assertion follows if we show that

(10.9) R1f∗(ΩY (logE)⊗OY OY (−dαf∗Ze)

)= 0,

where E = Z + F . Note that f∗Z = Z + 2F and we may assume that 0 < α 6 1.If 1/2 < α 6 1, then dαf∗Ze = f∗Z and (10.9) follows from [MP16, Th.B] using theprojection formula. On the other hand, if 0 < α 6 1/2, then dαf∗Ze = E and thevanishing follows from the fact that the pair (X,Z) is log canonical, using [GKKP11,Th. 14.1] (though, in this case, one could also check this directly).

Once we know that the filtration on M (hβ) is generated at level 0, it is straight-forward to check that

Ik(αZ) = mkx, for all α such that 0 < α 6 1 and all k > 0,

where mx is the ideal defining x in X.

Unlike in the case when D is a reduced integral divisor, when the filtrationF•OX(∗D) is generated at level n− 2 by [MP16, Th.B], in general it is not possibleto improve the bound given by Proposition 10.1.

Example 10.3 (Optimal generation level). — It can happen that on a surface X thefiltration on M (hβ) is not generated at level 0. Suppose, for example, that X = A2

and Z = L1 + L2 + L3, where L1, L2, and L3 are 3 lines passing through the ori-gin. If f : Y → X is the blow-up of the origin and E = (f∗Z)red, then we writeE = F +G1 +G2 +G3, where F is the exceptional divisor and the Gi are the stricttransforms of the Li. Let D = αZ with 0 < α 1, so that df∗De = E. If

H1(Y,ΩY (logE)⊗OY OY (−df∗De)

)= H1

(Y,ΩY (logE)⊗OY OY (−E)

)were zero, then it would follow from the standard exact sequence

0 −→ ΩY (logE)⊗OY OY (−E) −→ ΩY −→ ΩF ⊕3⊕i=1

ΩGi −→ 0

that the map

H0(Y,ΩY ) −→ H0(F,ΩF )⊕3⊕i=1

H0(Gi,ΩGi)

J.É.P. — M., 2019, tome 6


is surjective. In particular, we would deduce that the map

H0(X,ΩX) −→3⊕i=1

H0(Li,ΩLi)

is surjective. It is an easy exercise to see that this is not the case. Note that the non-vanishing of H1

(Y,ΩY (logE)⊗OY OY (−E)

)is not inconsistent with the Steenbrink-

type vanishing in [GKKP11, Th. 14.1], since the pair (X,Z) is not log-canonical.

Example 10.4 (Quasi-homogeneous isolated singularities). — For the class of quasi-homogeneous isolated singularities (such as those in the examples above), the gener-ation level for the filtration on M (hβ) can be detected by the Bernstein-Sato poly-nomial. Before formulating this more precisely, we recall some definitions. Supposethat Z is a hypersurface in X defined by h ∈ OX(X). The Bernstein-Sato polynomialof Z is the non-zero monic polynomial bh ∈ C[s] of smallest degree such that welocally have a relation of the form

bh(s)hs = P (s)•hs+1

for some nonzero P ∈ DX [s]. If Z is non-empty, it is known that (s + 1) divides bh;moreover, all the roots of bh are negative rational numbers. In this case, one definesαh = −λ, where λ is the largest root of the reduced Bernstein-Sato polynomial bh =

bh(s)/(s + 1). Note that bh has degree 0 if and only if Z is smooth, and in this caseone makes the convention that αh =∞.

The statement is that if Z = div(h) is reduced and has a unique singular point at x,which is a quasi-homogeneous singularity, and D = αZ, then the generation level k0of the filtration on M (hβ) (i.e., the smallest k such that the filtration is generated atlevel k) is

k0 = bn− αh − αc.This was proved by Saito [Sai09, Th. 0.7] when D is reduced, i.e., for α = 1, and wasextended to the general case by Zhang [Zha18].(4)

Note that for such singularities there is an explicit formula for αh; see e.g. [Sai09,§4.1]. Just as an illustration, for h = xy(x+y), which describes the previous example,we have αh = 2/3, and so for α small (more precisely 0 < α 6 1/3) we recover thefact that the generation level is equal to 1.

Example 10.5 (Incomensurability of higher Hodge ideals). — Suppose that X is asmooth surface and Z =

∑ri=1Di is a reduced effective divisor on X. Let f : Y → X

be a log resolution of (X,Z) that is an isomorphism overXrZ, and put E = (f∗Z)red.Let D =

∑ri=1(1−ai)Di be a divisor with 0 6 ai 1 for all i, so that df∗De = f∗Z.

In this case we have

R1f∗(OY (−df∗De)⊗ Ω1

Y (logE))

= 0

(4)Moreover, based on calculations of Saito, Zhang shows in loc. cit. that all Hodge ideals ofQ-divisors associated to such singularities can be computed explicitly.

J.É.P. — M., 2019, tome 6


by the projection formula and [MP16, Th.B], and so the filtration is generated atlevel 0. It follows from the discussion at the beginning of the section (see (10.3) and(10.4)) that if g is a local equation of Z, and D = αZ, with α 6 1 and close to 1,then Ik+1(D) is generated by g · Ik(D) and

h ·Q(w)− (α+ k)w ·Q(h) | w ∈ Ik(D), Q ∈ DerC(OX) .

For example, if X = C2 and Z is the cusp defined by x2 + y3, then for D = αZ

with α 6 1 and close to 1 we have

I0(D) = (x, y), I1(D) = (x2, xy, y3), and I2(D) = (x3, x2y2, xy3, y4− (2α+ 1)x2y).

Note in particular that if D1 = α1Z and D2 = α2Z, with α1 < α2 both close to 1,then there is no inclusion between the ideals I2(D1) and I2(D2). This is in contrastwith the picture for multiplier ideals, where for any Q-divisors D1 6 D2 one hasI0(D2) ⊆ I0(D1); see [Laz04, Prop. 9.2.32(i)]. It is not hard to check however that

I2(D1) = I2(D2) mod x2 + y3,

and that this is part of a general phenomenon where the picture is well behavedafter modding out by a defining equation for the hypersurface; this follows from theconnection with the V -filtration, see [MP18a, Cor. B].

Remark 10.6. — If the filtration is generated at level k, then Ik+1(D) is generatedby the terms appearing on the left hand side of conditions (10.3) and (10.4). A simplecalculation shows then that in this case, for every j > 1 and every x ∈ X, we have

multx Ik+j(D) > multx Ik+j−1(D) + multx Z − 1.

In particular, we have

multx Ik+j(D) > multx Ik(D) + j · (multx Z − 1).

Since the filtration is always generated at level n− 1 by Proposition 10.1, we obtainthe following consequence.

Corollary 10.7. — If D is an effective Q-divisor on the smooth variety X, withsupport Z, and if Z is singular at some x ∈ X, then Ij(D)x 6= OX,x for all j > n. Infact, if m = multx Z, then

multx Ij(D) > (j − n+ 1)(m− 1) for all j > n.

11. Non-triviality criteria. — The following is the analogue of [MP16, Th. 18.1] inthe setting of Q-divisors. Let D be an effective Q-divisor on the smooth variety X,with Z = Supp(D), and let ϕ : X1 → X be a projective morphism with X1 smooth,such that ϕ is an isomorphism over X r Z. We denote

Z1 = (ϕ∗Z)red and TX1/X = Coker(TX1−→ ϕ∗TX).

Theorem 11.1. — With the above notation, the following hold:(i) We have an inclusion

ϕ∗(Ik(ϕ∗D)⊗OX1

OX1(KX1/X + k(Z1 − ϕ∗Z))

)⊆ Ik(D).

J.É.P. — M., 2019, tome 6


(ii) If J is a coherent ideal on X such that J · TX1/X = 0, then

Jk · Ik(D) ⊆ ϕ∗(Ik(ϕ∗D)⊗OX1

OX1(KX1/X + k(Z1 − ϕ∗Z))).

Proof. — We may assume that D = α · div(h), for some α ∈ Q>0 and some nonzeroh ∈ OX(X). Let ψ : Y → X1 be a log resolution of (X1, ϕ

∗D) that is an isomorphismover X1 r ϕ−1(Z). We put

f = ϕ ψ and E = (f∗Z)red.

With the notation in Section 6, consider the filtered complex C• = C•g−α(−df∗De),where g = h f . We have an inclusion of complexes

A• = C• ⊗DY DY→X1−→ B• = C• ⊗DY DY→X .

Note that this is an injection due to the fact that OY (−df∗De) and ΩqY (logE) arelocally free sheaves of OY -modules, while all the maps FpDY→X1

→ FpDY→X aregenerically injective morphisms of locally free OY -modules. Consider, for any inte-ger k, the short exact sequence of complexes

0 −→ Fk−nA• −→ Fk−nB

• −→M• −→ 0.

Applying Rf∗ and taking the corresponding long exact sequence, we obtain a shortexact sequence

R0f∗Fk−nA• ι−−→ R0f∗Fk−nB

• −→ R0f∗M•.

If β = 1− α, it follows from Theorem 8.1 that

Rf∗Fk−nB• = R0f∗Fk−nB

• ' hβOX(KX + kZ +H

)⊗OX Ik(D)

and

Rg∗Fk−nA• = R0g∗Fk−nA

• ' hβOX1

(KX1

+ kZ1 + ϕ∗H)⊗OX1

Ik(ϕ∗D).

Therefore, after tensoring by OX(−H), the map ι induces a map

(11.1) ϕ∗(Ik(ϕ∗D)⊗ OX1(KX1 + kZ1)

)−→ Ik(D)⊗OX OX

(KX + kZ

).

Finally, the map ι is compatible with restriction to open subsets of X. By restrictingto an open subset X0 in the complement of Z, such that f is an isomorphism over X0,we see that the map in (11.1) is the identity on ωX0 . We thus deduce the assertionin (i) by tensoring (11.1) with OX

(−KX−kZ

). Furthermore, we see that the assertion

in (ii) follows if we show that Jk ·R0f∗M• = 0. Since

Mp = OY (−df∗De)⊗OY Ωn+pY (logE)⊗OY ψ∗(ϕ∗Fk+pDX/Fk+pDX1),

it is enough to show that under our assumption we have

ϕ∗FjDX · Jj ⊆ FjDX1for all j > 0.

This is proved in [MP16, Lem. 18.6].

We first use Theorem 11.1 in order to give a triviality criterion for Hodge ideals interms of invariants of a fixed resolution of singularities. We use this in turn in orderto bound the largest root of the reduced Bernstein-Sato polynomial (i.e., αh definedin Example 10.4) in terms of such invariants, in [MP18a, Cor.D].

J.É.P. — M., 2019, tome 6


Proposition 11.2. — Let Z be a reduced divisor on the smooth variety X, and letD = αZ, with α ∈ Q>0. Let f : Y → X be a log resolution of (X,Z) that is anisomorphism over X r Z and such that the strict transform Z of Z is smooth. Wedefine integers ai and bi by the expressions

f∗Z = Z +

m∑i=1

aiFi and KY/X =

m∑i=1

biFi,

where F1, . . . , Fm are the prime exceptional divisors. If

(11.2) bi + 1

ai> k + α for i such that 1 6 i 6 m,

then Ik(D) = OX((1− dαe)Z

). In particular, if 0 < α 6 1, then Ik(D) = OX .

Proof. — If D′ = α′Z, where α′ = α+ 1− dαe, then it follows from Lemma 4.4 thatIk(D) = Ik(D′) ⊗ OX

((1 − dαe)Z

). Since the inequalities (11.2) clearly also hold if

we replace α by α′, it follows that it is enough to treat the case 0 < α 6 1.First, note that since f∗D has simple normal crossings, by Proposition 7.1 we have

Ik(f∗D) = Ik(E)⊗ OY( m∑i=1

(1− dαaie)Fi),

where E = (f∗Z)red = Z+∑mi=1 Fi. We apply Theorem 11.1 (i) to obtain the inclusion

(11.3) f∗(Ik(E)⊗ OY (F )

)−→ Ik(D),

where

F :=

m∑i=1

(bi + k + 1− kai − dαaie

)Fi.

On the other hand, since E = Z +∑mi=1 Fi has simple normal crossings and Z is

smooth, it follows from the description of Hodge ideals of simple normal crossingdivisors in [MP16, Prop. 8.2] that we have

OY (−k ·m∑i=1

Fi) ⊆ Ik(E).

Note that the inequalities in (11.2) imply bi + 1 > kai + dαaie for all i, hence thedivisor F − k ·

∑mi=1 Fi is effective We thus deduce using (11.3) that we have

OX = f∗OY −→ Ik(D).

Remark 11.3. — More generally, suppose that we write Z =∑rj=1 Zj , and consider

an effective Q-divisor D =∑rj=1 αjZj supported on Z. For simplicity, let us assume

that 0 < αj 6 1 for all j. If f is a log resolution as in Proposition 11.2, and we write

f∗Zj = Zj +

m∑i=1

ajiFi

J.É.P. — M., 2019, tome 6


for all j (so that ai =∑rj=1 a

ji ), then the same proof gives Ik(D) = OX if

bi + 1 > kai +

m∑i=1

αjaji for all i.

We now turn our attention to non-triviality criteria for the Hodge ideals Ik(D) interms of the multiplicity of D, and of its support Z, along a given subvariety.

Corollary 11.4. — Let D be an effective Q-divisor on the smooth variety X, andlet Z be the support of D. If W is an irreducible closed subset of X of codimension rsuch that multW Z = a and multW D = b, and if q is a non-negative integer such that

b+ ka > q + r + 2k − 1,

then Ik(D) ⊆ I(q)W , the q-th symbolic power of IW . In particular, if

multW D >q + r + 2k − 1

k + 1,

then Ik(D) ⊆ I(q)W .

Proof. — After possibly restricting to a suitable open subset ofX meetingW , we mayassume that W is smooth. The first assertion in the corollary follows by applying11.1 (ii) to the blow-up ϕ : X1 → X along W . Note that we may take J = IWby [MP16, Ex. 18.7], while Ik(ϕ∗D) ⊆ OX1(Z1 − dϕ∗De). The last assertion followsthanks to the fact that by assumption we have a > b.

Remark 11.5. — An interesting consequence of the above corollary is that if Z is areduced divisor on the smooth, n-dimensional variety X, k is a positive integer, andx ∈ X is a point such that

multx Z > 2 +n

k,

then Ik(D) is non-trivial at x for every effective Q-divisor D with support Z (no mat-ter how small the coefficients).

Example 11.6 (Ordinary singularities, I). — Let X be a smooth variety of dimen-sion n, and Z a reduced divisor with an ordinary singularity at x ∈ X (recall thatthis means that the projectivized tangent cone of Z at x is smooth), for instance acone over a smooth hypersurface. If D = αZ, with α a rational number satisfying0 < α 6 1, then

multx Z 6n

k + α=⇒ Ik(D)x = OX,x.

Note that the converse of this statement will be proved in Corollary 11.8 below.Indeed, the assumption implies that after possibly replacing X by an open neigh-

borhood of x, the blow-up f : Y → X of X at x gives a log resolution of (X,Z).

J.É.P. — M., 2019, tome 6


Let E = F + Z, where F is the exceptional divisor of f and Z is the strict transformof Z. If m = multx Z, then we deduce from Theorem 11.1 that

f∗(Ik(ϕ∗D)⊗OY OY (KY/X + k(E − f∗Z))

)= f∗

(Ik(ϕ∗D)⊗OY OY ((n− 1 + k − km)F )

)⊆ Ik(D).

Now since ϕ∗D is supported on the simple normal crossings divisor E, by Proposi-tion 7.1 we have

Ik(ϕ∗D) = Ik(E)⊗OY OY ((1− dαme)F ),

where we use the fact that dαe = 1. Moreover, by [MP16, Prop. 8.2] we have

Ik(E) =(OY (−Z) + OY (−F )

)k ⊇ OY (−kF ).

Now by assumptionn− km− dαme > 0,

hence we deduce Ik(D) = OX .

Example 11.7 ( Ordinary singularities, II). — With considerable extra work, one cansay more in the ordinary case. We keep the notation of the previous example, andassume that x is a singular point of Z, hence m > 2. If k is a positive integer suchthat

(k − 1)m+ dαme < n and k 6 n− 2,

then we haveIk(D) = mkm+dαme−n

x

in a neighborhood of x, where mx is the ideal defining x (with the convention thatmjx=OX if j60). The argument is similar to that in [MP16, Prop. 20.7], so we omit it.

In what follows we make use of some general properties of Hodge ideals that willbe proved in Ch.D, namely the Restriction and Semicontinuity Theorems.

Corollary 11.8. — If X is a smooth n-dimensional variety, Z is a reduced divisorwith an ordinary singularity of multiplicity m > 2 at x ∈ X, and D = αZ with0 < α 6 1, then

Ik(D)x = OX,x ⇐⇒ m 6n

k + α.

Proof. — The “if” part follows directly from Example 11.6. For the converse, weneed to show that if mx is the ideal defining x and m > n/(k + α), then Ik(D) ⊆ mx.We may assume that Z is defined in X by h ∈ OX(X). Let r > 0 be such thatn+r = mk+dmαe−1 and consider the divisor Z ′ inX×Cr defined by h+ym1 +· · ·+ymr ,where y1, . . . , yr are the coordinates on Cr. It is easy to check that Z ′ is reduced andhas an ordinary singularity at (x, 0). By the Restriction Theorem (see Theorem 13.1and Remark 13.4 below), we have Ik(αZ) ⊆ Ik(αZ ′) · OX , where we consider Xembedded in X × Cr as X × 0. After replacing X and Z by X ′ and Z ′, we may

J.É.P. — M., 2019, tome 6


thus assume that n = mk+ dmαe− 1. If k 6 n− 2, then we may apply Example 11.7to conclude that Ik(D) ⊆ mx. Otherwise we have

k > n− 1 = mk + dmαe − 2,

which easily implies m = 2, k = 1, and α 6 12 , hence n = 2. Since Z has an ordinary

singularity at x, it follows that it must be a node, and in this case we have I1(αZ) = mxby Example 10.2.

Remark 11.9. — One can give an alternative argument, arguing as follows. Supposethat Z is a reduced divisor in X, defined by h ∈ OX(X). It is shown in [MP18a,Cor. C] that for 0 < α 6 1, we have Ik(αZ) = OX if and only if k 6 αh − α. If Z hasan ordinary singularity at x ∈ X, of multiplicity m > 2, then after replacing X by asuitable neighborhood of x, we have αh = n/m (see [Sai16, §2.5]), and we recover theassertion in Corollary 11.8.

Question 11.10. — Is it true that if X is a smooth n-dimensional variety, Z is areduced divisor on X, D is an effective Q-divisor with support Z, and for a pointx ∈ Zsing we have

k ·multx Z + multxD > n,

then Ik(D) ⊆ mx?

This would be a natural improvement of Corollary 11.4, and it does hold when D isreduced by [MP16, Cor. 21.3]. We may of course assume that dDe = Z, since otherwisethe inclusion is trivial (see Remark 4.3). At the moment we have:

Corollary 11.11. — Question 11.10 has a positive answer if D is of the form D=αZ.

Proof. — We may assume that α 6 1 and, arguing as in the proof of [MP16, Th. E],we construct a reduced divisor F on X×U , for a smooth variety U , such that for t ∈ Ugeneral the divisor Ft = F |X×t is reduced, with an ordinary singularity at (x, t) ofmultiplicity m = multx Z, and for some t0 ∈ U , the isomorphism X ' X × t0maps D to Ft0 . In this case Corollary 11.8 implies that Ik(Ft) vanishes at (x, t) fort ∈ U general, and the Semicontinuity Theorem (see Theorem 14.1 below) impliesthat Ik(Ft0) vanishes at (x, t0).

This allows us in particular to provide an analogue of [MP16, Th.A]:

Corollary 11.12. — If D is of the form D = αZ, then

Z is smooth ⇐⇒ Ik(D) = OX(Z − dDe) for all k.

Proof. — It suffices to assume 0 < α 6 1, in which case the condition becomesIk(D) = OX for all k. By Corollary 11.11 however, if multx Z > 2, then Ik(D) ⊆ mxfor all k > (n/2)− α.

J.É.P. — M., 2019, tome 6


12. Vanishing theorem. — As usual, we consider an effective Q-divisor D with sup-port Z, on the smooth variety X. In this section we assume that X is projective, andprove a vanishing theorem for Hodge ideals, extending [MP16, Th. F] as well as NadelVanishing for Q-divisors.

We start by choosing a positive integer ` such that `D is an integral divisor, andfurther assume that there exists a line bundle M on X such that

(12.1) M⊗` ' OX(`D),

so that the setting of Section 5 applies. We note that this can always be achievedafter passing to a finite flat cover of X.

Theorem 12.1. — Let X be a smooth projective variety of dimension n and D aneffective Q-divisor on X such that (12.1) is satisfied. Let L be a line bundle on X

such that L+Z−D is ample. For some k > 0, assume that the pair (X,D) is reduced(k − 1)-log-canonical, i.e., I0(D) = · · · = Ik−1(D) = OX(Z − dDe).(5) Then we have:

(1) If k 6 n, and L(pZ − dDe) is ample for all p such that 2 6 p 6 k + 1, then

Hi(X,ωX ⊗ L((k + 1)Z)⊗ Ik(D)

)= 0

for all i > 2. Moreover,

H1(X,ωX ⊗ L((k + 1)Z)⊗ Ik(D)

)= 0

holds if Hj(X,Ωn−jX ⊗ L((k − j + 2)Z − dDe)

)= 0 for all j such that 1 6 j 6 k.

(2) If k > n + 1, then Z must be smooth by Corollary 10.7, and so Ik(D) =

OX(Z − dDe) by Corollary 3.2. In this case, if L is a line bundle such thatL((k + 1)Z − dDe) is ample, then

Hi(X,ωX ⊗ L((k + 1)Z)⊗ Ik(D)

)= 0 for all i > 0.

(3) If U = X r Z is affine (e.g. if D or Z are ample), then (1) and (2) also holdwith L = M(−Z), assuming that M(pZ −dDe) is ample for p such that 1 6 p 6 k.(6)

Proof. — We use the notation in Section 5 and Remark 4.3. In particular, we considerthe filtered left DX -module

M1 = M ⊗OX OX(∗Z),

which we know is a direct summand in a filtered D-module underlying a mixed Hodgemodule on X. Its filtration satisfies

FkM1 'M(−Z)⊗ OX((k + 2)Z − dDe

)⊗ I ′k(D).

Note also that since L + Z −D is ample, there exists an ample line bundle A on Xsuch that L 'M(−Z)⊗A.

(5)Recall from Definition 9.3 that equivalently this means I′0(D) = · · · = I′k−1(D) = OX . By con-vention the condition is vacuous when k = 0.

(6)When k > 1, the condition of U being affine is in fact implied by the positivity condition, sinceD + Z − dDe is then an ample divisor with support Z.

J.É.P. — M., 2019, tome 6


Let us prove (1), i.e., let us consider the case k 6 n. The statement is equivalentto the vanishing of the cohomology groups

Hi(X,ωX ⊗ L((k + 2)Z − dDe)⊗ I ′k(D)

)= 0

Since I ′k−1(D) = OX , we have a short exact sequence

0 −→ ωX ⊗ L((k + 1)Z − dDe) −→ ωX ⊗ L((k + 2)Z − dDe)⊗ I ′k(D)

−→ ωX ⊗A⊗ grFk M1 −→ 0.

By taking the corresponding long exact sequence in cohomology and using Kodairavanishing, we see that the vanishing we are aiming for is equivalent to the samestatement for

Hi(X,ωX ⊗A⊗ grFk M1

).

We now consider the complex

C• :=(grF−n+k DR(M1)⊗A

)[−k].

Given the hypothesis on the ideals I ′p(D), this can be identified with a complex of theform[Ωn−kX ⊗ L(2Z − dDe) −→ Ωn−k+1

X ⊗ L⊗ OZ(3Z − dDe)

−→ · · · −→ Ωn−1X ⊗ L⊗ OZ((k + 1)Z − dDe

)−→ ωX ⊗A⊗ grFk M1

]placed in degrees 0 up to k. Saito’s Vanishing theorem [Sai90, §2.g] gives

(12.2) Hj(X,C•) = 0 for all j > k + 1.

We use the spectral sequence

Ep,q1 = Hq(X,Cp) =⇒ Hp+q(X,C•).

The vanishing statements we are interested in are for the terms Ek,i1 with i > 1. Wewill in fact show that

(12.3) Ek,ir = Ek,ir+1, for all r > 1.

This implies thatEk,i1 = Ek,i∞ = 0,

where the vanishing follows from (12.2) since i > 1, and this gives our conclusion.We are thus left with proving (12.3). On one hand we always have Ek+r,i−r+1

r = 0

because Ck+r = 0. On the other hand, we will show that under our hypothesis wehave Ek−r,i+r−11 = 0, from which we infer that Ek−r,i+r−1r = 0 as well, allowing usto conclude. To this end, note first that if r > k this vanishing is clear, since thecomplex C• starts in degree 0. If k = r, we have

E0,i+k−11 = Hi+k−1(X,Ωn−kX ⊗ L(2Z − dDe)

).

If i > 2 this is 0 by Nakano vanishing, while if i = 1 it is 0 because of our hypothesis.Finally, if k > r + 1, we have

Ek−r,i+r−11 = Hi+r−1(X,Ωn−rX ⊗ L⊗ OZ((k − r + 2)Z − dDe)),

J.É.P. — M., 2019, tome 6


which sits in an exact sequence

Hi+r−1(X,Ωn−rX ⊗ L((k − r + 2)Z − dDe))−→ Ek−r,i−r+1

1

−→ Hi+r(X,Ωn−rX ⊗ L((k − r + 1)Z − dDe)

).

We again have two cases:(1) If i > 2, we deduce that Ek−r,i+r−11 = 0 by Nakano vanishing.(2) If i = 1, using Nakano vanishing we obtain a surjective morphism

Hr(X,Ωn−rX ⊗ L((k − r + 2)Z − dDe)

)−→ Ek−r,i+r−11 ,

and if the extra hypothesis on the term on the left holds, then we draw the sameconclusion as in (1).

The same argument proves (3), once we replace Saito Vanishing (12.2) by thevanishing

Hi(X, grFk DR(M1)

)= 0

for all i > 0 and all k, which in turn is implied by the same statement for theDX -module M underlying a Hodge D-module, in which M1 is a direct summand.Furthermore, this is implied by the vanishing of the perverse sheaf cohomology

Hi(X,DR(M )

)= 0 for all i > 0.

Indeed, by the strictness property for direct images (see e.g. [MP16, Ex. 4.2]), for(M , F ) we have the decomposition

Hi(X,DR(M )

)'⊕q∈Z

Hi(X, grF−q DR(M )

).

Recall now from Section 5 that M ' j+N , where N underlies a Hodge D-moduleon U , and j : U → X is the inclusion. Denoting P = DR(M ), we then have P 'j∗j∗P , and so it suffices to show that

Hi(U, j∗P ) = 0 for all i > 0.

But this is a consequence of Artin vanishing (see e.g. [Dim04, Cor. 5.2.18]), since U isaffine.

Finally, the assertion in (2) follows from Kodaira vanishing, using the long exactsequence in cohomology associated to the short exact sequence

0 −→ ωX ⊗ L((k + 1)Z − dDe

)−→ ωX ⊗ L

((k + 2)Z − dDe

)−→ ωZ ⊗ L

((k + 1)Z − dDe

)|Z −→ 0.

Remark 12.2. — We expect the statement of the theorem to hold even without as-suming the existence of M (i.e., of an `-th root of the line bundle OX(`D)). Thisis known for k = 0, when the statement follows from Nadel Vanishing, see [Laz04,Th. 9.4.8]. However, at the moment we do not know how to show this for k > 1.

J.É.P. — M., 2019, tome 6


Remark 12.3 (Toric varieties). — As in [MP16, Cor. 25.1], when X is a toric varietythe Nakano-type vanishing requirement in Theorem 12.1 (1) is automatically satisfiedthanks to the Bott-Danilov-Steenbrink vanishing theorem. A stronger result in thissetting is proved in [Dut18].

Remark 12.4 (Projective space, abelian varieties). — As in [MP16, Th. 25.3 & 28.2],appropriate statements on Pn and abelian varieties work without the extra assump-tions of reduced log canonicity and Nakano-type vanishing in Theorem 12.1. Moreprecisely, keeping the notation at the beginning of the section, we have:

Variant 12.5. — Let D be an effective Q-divisor on Pn which is numerically equiv-alent to a hypersurface of degree d > 1. If ` > d− n− 1, then

Hi(Pn,OPn(`)⊗ OPn(kZ)⊗ Ik(D)

)= 0 for all i > 0.

Note that the positivity condition in Theorem 12.1 is satisfied, since for everyeffective Q-divisor D 6= 0 in Pn we have degdDe < degD + degZ.

Variant 12.6. — If X is an abelian variety and D is an ample Q-divisor on X, then

Hi(X,M(kZ)⊗ Ik(D)⊗ α) = 0

for all i > 0 and α ∈ Pic0(X).

Note that on an abelian variety every effective Q-divisor is nef, and the amplenessof D is equivalent to that of any divisor whose support is equal to that of D.

The proofs are completely similar to those in loc. cit., replacing OX(∗D) in thereduced case by M1 in the proof above, and noting that since M1 is a filtered directsummand in j+p+OV as in Section 5, the vanishing properties we use continue tohold.

D. Restriction, subadditivity, and semicontinuity theorems

In this part of the paper we provide Q-divisor analogues of the results in [MP18b].This extends well-known statements in the setting of multiplier ideals; further discus-sion and references regarding these can be found in loc. cit.

13. Restriction theorem. — We begin with the Q-divisor version of the RestrictionTheorem:

Theorem 13.1. — Let D be an effective Q-divisor, with support Z, on the smoothvariety X, and let Y be a smooth irreducible divisor on X such that Y 6⊆ Z. If wedenote DY = D|Y , ZY = Z|Y , and Z ′Y = (ZY )red, then for every k > 0 we have

(13.1) OY(−k(ZY − Z ′Y )

)· Ik(DY ) ⊆ Ik(D) · OY .

In particular, if ZY is reduced, then for every k > 0 we have

(13.2) Ik(DY ) ⊆ Ik(D) · OY .

Moreover, if Y is sufficiently general (e.g. a general member of a basepoint-free linearsystem), then we have equality in (13.2).

J.É.P. — M., 2019, tome 6


Remark 13.2. — Note that when D is a reduced divisor we have DY = ZY , andDY − Z ′Y is an integral divisor with support in Z ′Y . Therefore Lemma 4.4 gives

Ik(DY ) = OX(−(DY − Z ′Y )

)· Ik(Z ′Y ),

hence the statement in the theorem coincides with that of [MP18b, Th.A].

Proof of Theorem 13.1. — The argument follows the proof of [MP18b, Th.A], with asimplification observed in [Sai16], hence we only give the outline of the proof. Since thestatement is local, we may assume that D = α · div(h) for some nonzero h ∈ OX(X).Consider the following commutative diagram with Cartesian squares:

VYi′′ //

p′

V

p

UYi′ //

j′

U

j

Yi // X,

where p and j are as in diagram (2.3), while i is the inclusion of Y in X. Note thatif n = dim(X), we have a canonical base-change isomorphism

i!(j p)+QHV [n] ' (j′ p′)+i′′!QH

V [n]

proved in [Sai90, 4.4.3]. We also have a canonical isomorphism

i′′!QHV [n] = (QH

VY [n− 1])(−1)[−1]

(see for instance [Sai88, §3.5]). Here we use the Tate twist notation, which for a mixedHodge module M = (M , F•M ,K) is given by

M(k) =(M , F•−kM ,K ⊗Q Q(k)

).

We obtain, in particular, an isomorphism of filtered right DX -modules(H 1i!Mr(h

−α), F•)'(Mr(h|−αY ), F•+1

).

Recall now that if (VαM )α∈Q is the V -filtration on M = Mr(h−α) corresponding

to the smooth hypersurface Y ⊆ X, then there is a canonical morphism

σ : grV0 M −→ grV−1 M ⊗OX OX(Y )

such thatH 1i!M ' Coker(σ),

with the Hodge filtration on the right-hand side induced by the Hodge filtration on M .We refer to [MP18b, §2] for details.

One defines a morphism

η : Fk grV−1 M =FkV−1M

FkV<−1M−→ FkM ⊗OX OY

J.É.P. — M., 2019, tome 6


that maps the class of u ∈ FkV−1M = FkM∩V−1M to the class of u in FkM⊗OXOY .After tensoring η with OX(Y ), the resulting morphism vanishes on the image of therestriction of σ to Fk grV0 M , hence we obtain an induced morphism

(13.3) Fk+1Mr(h|−αY ) ' FkH 1i!M ' Fk Coker(σ) −→ FkM ⊗OX OY (Y ).

Applying this with k replaced by k− n, it follows from the definition of Hodge idealsand the formula for the equivalence between left and right D-modules that we havea morphism

Ik(DY )⊗OY ωY(kZ ′Y + div(h|Y )

)−→ Ik(D)⊗OX ωX

(kZ + div(h)

)⊗OX OY (Y ).

By tensoring this with ω−1Y(−kZY −div(h|Y )

)and composing with the canonical map

Ik(D)⊗OX OY → Ik(D) · OY , we obtain a canonical morphism

ϕ : OY(−k(ZY − Z ′Y )

)⊗ Ik(DY ) −→ Ik(D) · OY .

Note that all constructions are compatible with restrictions to open subsets and whenrestricting to Z = X r U , the above morphism can be identified with the identitymap on OY . Therefore the morphism ϕ is compatible with the two inclusions in OY ,and we deduce the inclusion in (13.1).

Suppose now that Y is general, so that ZY = Z ′Y and Y is non-characteristic withrespect to M . For example, this condition holds if Y is transversal to the strata ina Whitney stratification of Z (see [DMST06, §2]); in particular, it holds if Y is ageneral member of a basepoint-free linear system. We may assume that Y is definedby a global equation t ∈ OX(X). In this case, it follows from [Sai88, Lem. 3.5.6] thatgrV0 M = 0 and grV−1 M = M ⊗OX OY . It is now straightforward to check that themorphism (13.3) is an isomorphism, hence ϕ is an isomorphism, and we thus haveequality in (13.2).

We deduce the following analogue of inversion of adjunction:

Corollary 13.3. — With the notation of Theorem 13.1, if ZY is reduced andIk(DY )x = OY,x for some x ∈ Y , then Ik(D)x = OX,x.

Remark 13.4. — If D is an effective Q-divisor, with support Z, on the smooth vari-ety X, and Y is a smooth subvariety of X such that Y 6⊆ Z and Z|Y is reduced, thenfor every k > 0 we have

Ik(D|Y ) ⊆ Ik(D) · OY .This follows by writing Y locally as a transverse intersection of r smooth divisorson X and applying repeatedly the inclusion (13.2).

Remark 13.5. — With the notation in Theorem 13.1, let Y1, . . . , Yr be generalelements in a basepoint-free linear system on X, where r 6 n = dim(X). IfW = Y1 ∩ · · · ∩ Yr, then for every k > 0 we have

Ik(D|W ) = Ik(D) · OW .

Indeed, if Wi = Y1 ∩ · · · ∩ Yi, and if (Sβ)β are the strata of a Whitney stratificationof Z, then it follows by induction on i that we have a Whitney stratification of Z|Wi

J.É.P. — M., 2019, tome 6


with strata (Sβ ∩Wi)β . Moreover, Yi+1 is transversal to each such stratum. We maythus apply the theorem to each divisorD|Wi and smooth hypersurface Yi+1∩Wi ⊆Wi,to conclude that

Ik(D|W ) = Ik(D) · OW .

14. Semicontinuity theorem. — The same argument as in [MP18b, §5], based onthe Restriction Theorem (in this case Theorem 13.1 above), gives the following semi-continuity statement. The set-up is as follows: let f : X → T be a smooth morphismof relative dimension n between arbitrary varieties X and T , and s : T → X a mor-phism such that f s = IdT . Let D be an effective Q-Cartier Q-divisor on X, relativeover T (that is, we can write D locally as αH, for an effective divisor H and a positiverational number α, with H flat over T ). We assume that we have an effective divisor Zon X, relative over T , with Supp(Z) = Supp(D), and such that for every t ∈ T , therestriction Zt to the fiber Xt = f−1(t) is reduced. For every x ∈ X, we denote by mxthe ideal defining x in Xf(x).

Theorem 14.1. — With the above notation, for every q > 1, the set

Vq :=t ∈ T | Ik(Dt) 6⊆ mqs(t)

is open in T .

15. Subadditivity theorem. — The calculation for I2 in Example 10.5 shows thatthe inclusion

Ik(D1 +D2) ⊆ Ik(D1)

cannot hold for arbitrary Q-divisors D1 and D2. However, with an appropriate as-sumption on the support, we have the following stronger subadditivity statement:

Theorem 15.1. — If D1 and D2 are effective Q-divisors on the smooth variety X,whose supports Z1 and Z2 satisfy the property that Z1 +Z2 is reduced, then for everyk > 0 we have

Ik(D1 +D2) ⊆∑i+j=k

Ii(D1) · Ij(D2) · OX(−jZ1 − iZ2) ⊆ Ik(D1) · Ik(D2).

Note first that, for every i and j, the inclusion

FiM (h−α) ⊆ Fi+jM (h−α)

implies the inclusion

(15.1) OX(−jZ) · Ii(D) ⊆ Ii+j(D).

This gives the second inclusion in the statement above. To prove the first inclusion,as in the proof of [MP18b, Th.B] it is enough to show the following:(7)

(7)Indeed, the Restriction Theorem applies in the form given in Remark 13.4 for the diagonalembedding X → X ×X, since we are assuming that Z1 + Z2 is reduced.

J.É.P. — M., 2019, tome 6


Proposition 15.2. — Let X1 and X2 be smooth varieties and let Di be effectiveQ-divisors on Xi, with support Zi, for i = 1, 2. If Bi = p∗iDi, where pi : X1×X2 → Xi

are the canonical projections, then for every k > 0 we have

Ik(B1 +B2) =∑i+j=k

(Ii(D1)OX1

(−jZ1) · OX1×X2

)·(Ij(D2)OX2

(−iZ2) · OX1×X2

).

Proof. — By Remark 2.2, we can assume that there exist regular functions h1 on X1

and h2 on X2, together with α ∈ Q>0, such that Ii(D1) and Ij(D2) are defined byMr(h

−α1 ) and Mr(h

−α2 ), respectively. The statement follows precisely as in [MP18b,

Prop. 4.1], as long as we show that there is a canonical isomorphism of filteredD-modules (

Mr((p∗1h1 · p∗2h2)−α), F

)'(Mr(h

−α1 )Mr(h

−α2 ), F

),

where the filtration on the right hand side is the exterior product of the filtrationson the two factors. But this is a consequence of the canonical isomorphism of mixedHodge modules

j∗p∗QHV1×V2

[n1 + n2] ' j1∗p1∗QHV1

[n1] j2∗p2∗QHV2

[n2],

with the obvious notation as in (2.3) for i = 1, 2, together with Lemma 2.8.

References[Bjö93] J.-E. Björk – Analytic D-modules and applications, Mathematics and its Applications,

Kluwer Academic Publisher, Dordrecht, 1993.[Dim04] A. Dimca – Sheaves in topology, Universitext, Springer-Verlag, Berlin, New York, 2004.[DMST06] A. Dimca, Ph. Maisonobe, M. Saito & T. Torrelli – “Multiplier ideals, V -filtrations and

transversal sections”, Math. Ann. 336 (2006), no. 4, p. 901–924.[Dut18] Y. Dutta – “Vanishing for Hodge ideals on toric varieties”, arXiv:1807.11911, to appear

in Math. Nachr., 2018.[Ful93] W. Fulton – Introduction to toric varieties, Ann. of Math. Studies, vol. 131, Princeton

University Press, Princeton, NJ, 1993.[GKKP11] D. Greb, S. Kebekus, S. J. Kovács & T. Peternell – “Differential forms on log canonical

spaces”, Publ. Math. Inst. Hautes Études Sci. 114 (2011), p. 87–169.[HTT08] R. Hotta, K. Takeuchi & T. Tanisaki – D-modules, perverse sheaves, and representation

theory, Progress in Math., vol. 236, Birkhäuser, Boston, Basel, Berlin, 2008.[Laz04] R. Lazarsfeld – Positivity in algebraic geometry. II, Ergeb. Math. Grenzgeb. (3), vol. 49,

Springer-Verlag, Berlin, 2004.[MP16] M. Mustata & M. Popa – “Hodge ideals”, arXiv:1605.08088, to appear in Memoirs of the

AMS, 2016.[MP18a] , “Hodge ideals for Q-divisors, V -filtration, and minimal exponent”, arXiv:

1807.01935, 2018.[MP18b] , “Restriction, subadditivity, and semicontinuity theorems for Hodge ideals”, In-

ternat. Math. Res. Notices (2018), no. 11, p. 3587–3605.[Pop19] M. Popa – “D-modules in birational geometry”, in Proceedings of the ICM (Rio de Janeiro,

2018), Vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2019, p. 781–806.[Sai88] M. Saito – “Modules de Hodge polarisables”, Publ. RIMS, Kyoto Univ. 24 (1988), no. 6,

p. 849–995.[Sai90] , “Mixed Hodge modules”, Publ. RIMS, Kyoto Univ. 26 (1990), no. 2, p. 221–333.[Sai07] , “Direct image of logarithmic complexes and infinitesimal invariants of cycles”,

in Algebraic cycles and motives. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 344,Cambridge Univ. Press, Cambridge, 2007, p. 304–318.

J.É.P. — M., 2019, tome 6

http://arxiv.org/abs/1807.11911





[Sai09] , “On the Hodge filtration of Hodge modules”, Moscow Math. J. 9 (2009), no. 1,p. 161–191.

[Sai16] , “Hodge ideals and microlocal V -filtration”, arXiv:1612.08667, 2016.[Sai17] , “A young person’s guide to mixed Hodge modules”, in Hodge theory and L2-ana-

lysis, Adv. Lect. Math. (ALM), vol. 39, Int. Press, Somerville, MA, 2017, p. 517–553.[Zha18] M. Zhang – “Hodge filtration for Q-divisors with quasi-homogeneous isolated singulari-

ties”, arXiv:1810.06656, 2018.

Manuscript received 19th November 2018accepted 13th May 2019

Mircea Mustata, Department of Mathematics, University of MichiganAnn Arbor, MI 48109, USAE-mail : [email protected] : http://www-personal.umich.edu/~mmustata/

Mihnea Popa, Department of Mathematics, Northwestern University2033 Sheridan Road, Evanston, IL 60208, USAE-mail : [email protected] : https://sites.math.northwestern.edu/~mpopa/

J.É.P. — M., 2019, tome 6



mailto:[email protected]

http://www-personal.umich.edu/~mmustata/

mailto:[email protected]

https://sites.math.northwestern.edu/~mpopa/

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